From diana.mecum at gmail.com  Sun Jul  1 05:58:30 2007
From: diana.mecum at gmail.com (Diana Mecum)
Date: Sat, 30 Jun 2007 22:58:30 -0500
Subject: Question related to sequence A066452
Message-ID: <a851af150706302058k5519146fr4bdd069236671ae5@mail.gmail.com>

Folks,

I am trying to add extension terms for sequence A066452. I have a question.

This is the internal text describing the sequence:

%I A066452
%S A066452
1,1,2,1,4,1,4,4,3,2,8,3,7,7,9,2,8,5,10,10,8,6,19,6,12,9,9,8,22,9,12,
%T A066452 12,15,10,31,9,11,14,24,13,23,9,24,17,16,10,35,15,23
%N A066452 Anti-phi(n).
%H A066452 Jon Perry, <a href="
http://www.users.globalnet.co.uk/~perry/maths/antidivisorother2.htm">
               Anti-phi function</a>
%F A066452 anti-phi(n) = number of integers <= n that are coprime to the
anti-divisors of n
%e A066452 10 has the anti-divisors 3,4,7. Therefore numbers coprime to
3,4,7 and less than
               10 are are 1,2,5, therefore anti-phi(10)=3.
%Y A066452 Cf. A058838, A066241.
%Y A066452 Sequence in context: A024994 A051953 A079277 this_sequence
A007104 A102627 A088296
%Y A066452 Adjacent sequences: A066449 A066450 A066451 this_sequence A066453
A066454 A066455
%K A066452 nonn,more,easy
%O A066452 2,3
%A A066452 Jon Perry (perry(AT)globalnet.co.uk), Dec 29 2001

I found a definition for "anti-divisor" as follows:

"Non-divisor: a number k which does not divide a given number x."
"Anti-divisor: a non-divisor k of x with the property that k is an odd
divisor of 2x-1 or 2x+1, or an even divisor of 2x."

I see how Jon gets 3, 4 and 7 as anti-divisors of 10. However, 2 is not
coprime to the anti-divisors of 10. He has 1, 2, and 5 as on the anti-phi
list.

The sequence which I derived for this sequence is:

1, 1, 2, 1, 4, 1, 4, 4, 3, 2, 2, 5, 3, 5, 4, 9, 2, 4, 5, 6, 6, 6, 6, 10, 5,
8, 6, 5, 8, 8, 9, 12,
 7, 10, 7, 12, 9, 8, 9, 13, 13, 9, 9, 14, 10

Can someone tell me if I am misunderstanding the definition of the sequence,
or if I have found an error?

Thanks,

Diana M.

-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From joshua.zucker at gmail.com  Sun Jul  1 06:44:41 2007
From: joshua.zucker at gmail.com (Joshua Zucker)
Date: Sat, 30 Jun 2007 21:44:41 -0700
Subject: Question related to sequence A066452
In-Reply-To: <a851af150706302058k5519146fr4bdd069236671ae5@mail.gmail.com>
References: <a851af150706302058k5519146fr4bdd069236671ae5@mail.gmail.com>
Message-ID: <721e81490706302144v4a71e477jbca73f5e29577eea@mail.gmail.com>

I don't understand it either, but at least I could use the wayback
machine to track down the URL given, at
  http://tinyurl.com/2xuvgr

And it says
  The anti-phi function is defined as the numbers <n that do not have
any anti-divisor as a factor.

Which may or may not be what they actually mean ... but at least it's
another possible interpretation to try.

--Joshua Zucker


On 6/30/07, Diana Mecum <diana.mecum at gmail.com> wrote:
> Folks,
>
> I am trying to add extension terms for sequence A066452. I have a question.
>
> This is the internal text describing the sequence:
>
> %I A066452
> %S A066452
> 1,1,2,1,4,1,4,4,3,2,8,3,7,7,9,2,8,5,10,10,8,6,19,6,12,9,9,8,22,9,12,
> %T A066452
> 12,15,10,31,9,11,14,24,13,23,9,24,17,16,10,35,15,23
> %N A066452 Anti-phi(n).
> %H A066452 Jon Perry, <a
> href="http://www.users.globalnet.co.uk/~perry/maths/antidivisorother2.htm
> ">
>                Anti-phi function</a>
> %F A066452 anti-phi(n) = number of integers <= n that are coprime to the
> anti-divisors of n
> %e A066452 10 has the anti-divisors 3,4,7. Therefore numbers coprime to
> 3,4,7 and less than
>                10 are are 1,2,5, therefore anti-phi(10)=3.
> %Y A066452 Cf. A058838, A066241.
> %Y A066452 Sequence in context: A024994 A051953 A079277 this_sequence
> A007104 A102627 A088296
> %Y A066452 Adjacent sequences: A066449 A066450 A066451 this_sequence A066453
> A066454 A066455
> %K A066452 nonn,more,easy
> %O A066452 2,3
> %A A066452 Jon Perry (perry(AT)globalnet.co.uk), Dec 29 2001
>
> I found a definition for "anti-divisor" as follows:
>
> "Non-divisor: a number k which does not divide a given number x."
> "Anti-divisor: a non-divisor k of x with the property that k is an odd
> divisor of 2x-1 or 2x+1, or an even divisor of 2x."
>
> I see how Jon gets 3, 4 and 7 as anti-divisors of 10. However, 2 is not
> coprime to the anti-divisors of 10. He has 1, 2, and 5 as on the anti-phi
> list.
>
> The sequence which I derived for this sequence is:
>
> 1, 1, 2, 1, 4, 1, 4, 4, 3, 2, 2, 5, 3, 5, 4, 9, 2, 4, 5, 6, 6, 6, 6, 10, 5,
> 8, 6, 5, 8, 8, 9, 12,
>  7, 10, 7, 12, 9, 8, 9, 13, 13, 9, 9, 14, 10
>
> Can someone tell me if I am misunderstanding the definition of the sequence,
> or if I have found an error?
>
> Thanks,
>
> Diana M.
>
> --
> "God made the integers, all else is the work of man."
> L. Kronecker, Jahresber. DMV 2, S. 19.




From zbi74583 at boat.zero.ad.jp  Sun Jul  1 06:57:40 2007
From: zbi74583 at boat.zero.ad.jp (koh)
Date: Sun, 01 Jul 2007 13:57:40 +0900
Subject: Quiz
Message-ID: <20070701045738.zbi74583@boat.zero.ad.jp>

    Hi, Seqfans

    What are these sequences?

    S1 : 1,1,1,0,1,0,1,-1,0,1,2,2....
    S2 : 1,2,3,-2,0,3,1,2,1,1....

       Hint.... Graph, Euler Number, S1=English, S2=Japanese

    Yasutoshi
    




From reismann at free.fr  Sun Jul  1 11:52:37 2007
From: reismann at free.fr (reismann at free.fr)
Date: Sun, 01 Jul 2007 11:52:37 +0200
Subject: =?iso-8859-1?b?Qe5vbg==?= and Chronos
Message-ID: <1183283557.468779659d8ca@imp.free.fr>

Hi seqfans,

A?on and Chronos form the Deleuze's concept of time :
Chronos : pulsated time (beats of our heart), the time of the history, the time
which passes.
A?on : the pure moment of time, the time of the events, of the "ecc??t?s", a
time at the same time too late and too early.

Ordinal and cardinal function of the numbers :
There are apples on the table.
1, 2, 3, 4, 5, 6. I count apples, the numbers have an ordinal function, the
action to count is held in the Chronos.
There are thus 6 apples. 6 take a cardinal function, we are not located more in
the Chronos, we are in another time : the A?on.

In our vision of the natural numbers, in the vision of the fundamental theorem
of the arithmetic, in the decomposition in weight*level with jump=0, we are in
the A?on.
To replace the natural numbers in the Chronos, it is enough to consider the
decomposition in weight*level+jump with jump=1.
We did not see the difference until now because the two decompositions give the
same result (with a different offset). I think that it is thus because the jump
is constant, the pulsation is regular.

In A?on, the prime numbers are primes, they are of level 1, it is all.
How to see in a new way prime numbers ?
By considering them in the Chronos, by analyzing them by the decomposition in
weight*level+jump.
In this case the jump is not constant, the pulsation is irregular.
There are "prime numbers" and "multiples" among the prime numbers ("prime
numbers" and "multiples" remain to be defined in this case).

The numbers have only one A?on but several Chronos. It is what I wanted to say
by ?Numbers are nothings?. The numbers are nothing if one does not specify in
which time one is located. Is the number in the A?on or the Chronos ? And if it
is in the Chronos, in which Chronos is it ?
11 in A?on : 11 is prime, is of level 1.
11 in the Chronos of natural numbers : 11 has a weight of 2, 11 = 2*5+1.
11 in the Chronos of prime numbers : 11 has a weight of 3 : 11 = 3*3+2.

Good thoughts,

R?mi Eismann

Deleuze on Wikipedia en :
http://en.wikipedia.org/wiki/Deleuze
Deleuze on Wikipedia fr :
http://fr.wikipedia.org/wiki/Gilles_Deleuze
On A?on and chronos (in french) :
Le vocabulaire de Deleuze (r?alis? par Rapha?l Bessis) -
http://tuxcafe.org/~renee/textes/deleuze/vocabulaire_deleuze.pdf





From diana.mecum at gmail.com  Sun Jul  1 16:56:15 2007
From: diana.mecum at gmail.com (Diana Mecum)
Date: Sun, 1 Jul 2007 09:56:15 -0500
Subject: Question related to sequence A066452
In-Reply-To: <721e81490706302144v4a71e477jbca73f5e29577eea@mail.gmail.com>
References: <a851af150706302058k5519146fr4bdd069236671ae5@mail.gmail.com>
	 <721e81490706302144v4a71e477jbca73f5e29577eea@mail.gmail.com>
Message-ID: <a851af150707010756o381941eau786ba39833a11510@mail.gmail.com>

Joshua,

Thanks a bunch. I would never have been able to find this different spin on
the definition of anti-phi.

I have been able to replicate the original list.

Diana M.

On 6/30/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
>
> I don't understand it either, but at least I could use the wayback
> machine to track down the URL given, at
>   http://tinyurl.com/2xuvgr
>
> And it says
>   The anti-phi function is defined as the numbers <n that do not have
> any anti-divisor as a factor.
>
> Which may or may not be what they actually mean ... but at least it's
> another possible interpretation to try.
>
> --Joshua Zucker
>
>
> On 6/30/07, Diana Mecum <diana.mecum at gmail.com> wrote:
> > Folks,
> >
> > I am trying to add extension terms for sequence A066452. I have a
> question.
> >
> > This is the internal text describing the sequence:
> >
> > %I A066452
> > %S A066452
> > 1,1,2,1,4,1,4,4,3,2,8,3,7,7,9,2,8,5,10,10,8,6,19,6,12,9,9,8,22,9,12,
> > %T A066452
> > 12,15,10,31,9,11,14,24,13,23,9,24,17,16,10,35,15,23
> > %N A066452 Anti-phi(n).
> > %H A066452 Jon Perry, <a
> > href="
> http://www.users.globalnet.co.uk/~perry/maths/antidivisorother2.htm
> > ">
> >                Anti-phi function</a>
> > %F A066452 anti-phi(n) = number of integers <= n that are coprime to the
> > anti-divisors of n
> > %e A066452 10 has the anti-divisors 3,4,7. Therefore numbers coprime to
> > 3,4,7 and less than
> >                10 are are 1,2,5, therefore anti-phi(10)=3.
> > %Y A066452 Cf. A058838, A066241.
> > %Y A066452 Sequence in context: A024994 A051953 A079277 this_sequence
> > A007104 A102627 A088296
> > %Y A066452 Adjacent sequences: A066449 A066450 A066451 this_sequence
> A066453
> > A066454 A066455
> > %K A066452 nonn,more,easy
> > %O A066452 2,3
> > %A A066452 Jon Perry (perry(AT)globalnet.co.uk), Dec 29 2001
> >
> > I found a definition for "anti-divisor" as follows:
> >
> > "Non-divisor: a number k which does not divide a given number x."
> > "Anti-divisor: a non-divisor k of x with the property that k is an odd
> > divisor of 2x-1 or 2x+1, or an even divisor of 2x."
> >
> > I see how Jon gets 3, 4 and 7 as anti-divisors of 10. However, 2 is not
> > coprime to the anti-divisors of 10. He has 1, 2, and 5 as on the
> anti-phi
> > list.
> >
> > The sequence which I derived for this sequence is:
> >
> > 1, 1, 2, 1, 4, 1, 4, 4, 3, 2, 2, 5, 3, 5, 4, 9, 2, 4, 5, 6, 6, 6, 6, 10,
> 5,
> > 8, 6, 5, 8, 8, 9, 12,
> >  7, 10, 7, 12, 9, 8, 9, 13, 13, 9, 9, 14, 10
> >
> > Can someone tell me if I am misunderstanding the definition of the
> sequence,
> > or if I have found an error?
> >
> > Thanks,
> >
> > Diana M.
> >
> > --
> > "God made the integers, all else is the work of man."
> > L. Kronecker, Jahresber. DMV 2, S. 19.
>



-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From jvospost3 at gmail.com  Sun Jul  1 18:28:56 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Sun, 1 Jul 2007 09:28:56 -0700
Subject: =?WINDOWS-1252?Q?Re:_A=EEon_and_Chronos?=
In-Reply-To: <1183283557.468779659d8ca@imp.free.fr>
References: <1183283557.468779659d8ca@imp.free.fr>
Message-ID: <5542af940707010928h4890e4b4ofa519b20a082694@mail.gmail.com>

Drifting away from seqfans primary function as I understand it, but
directly on your topic, as I understand it:

There is profane time, and there is sacred time. According to Eliade,
myths describe a time that is fundamentally different from historical
time (what modern man would consider "normal" time). "In short," says
Eliade, "myths describe ? breakthroughs of the sacred (or the
'supernatural') into the World".[7] The mythical age is the time when
the Sacred entered our world, giving it form and meaning: "The
manifestation of the sacred ontologically founds the world".[8] Thus,
the mythical age is sacred time, the only time that has value for
traditional man.

[7] Mircea Eliade, Myth and Reality, pg. 6
[8] Mircea Eliade, The Sacred and the Profane, pg. 21

http://en.wikipedia.org/wiki/Eternal_return_(Eliade)





From reismann at free.fr  Sun Jul  1 19:05:19 2007
From: reismann at free.fr (reismann at free.fr)
Date: Sun, 01 Jul 2007 19:05:19 +0200
Subject: =?iso-8859-1?b?Qe5vbg==?= and Chronos
In-Reply-To: <5542af940707010928h4890e4b4ofa519b20a082694@mail.gmail.com>
References: <1183283557.468779659d8ca@imp.free.fr> <5542af940707010928h4890e4b4ofa519b20a082694@mail.gmail.com>
Message-ID: <1183309519.4687decf17576@imp.free.fr>

I agree, it is not the place to speak philosophy but as nobody speaks about my
mathematical work...

I do not agree on this concept of time. The sacred time, the mythical age, the
"jadis', the ?it was once? of the tales are not times. They are "ecc??t?s", they
make much more than to define a temporal framework, they create an environment,
an atmosphere. The time of these "ecc??t?s" is indeed A?on.

But the disctinction between A?on and Chronos is not the same one as between the
sacred time and the profane time. Not. A?on is the pure moment of time, it is
the point on the arrow of time, it is the time of a photography. In mathematics
and sciences in general, one uses A?on, one analyzes photographs. By putting the
numbers in their Chronos, their fitting, their otherness, we see them in another
manner.

With the decomposition of the prime numbers in weight*level+gap, I replace the
prime numbers in their Chronos.

Best,

R?mi Eismann




From jvospost3 at gmail.com  Sun Jul  1 19:18:16 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Sun, 1 Jul 2007 10:18:16 -0700
Subject: =?ISO-8859-1?Q?Re:_A=EEon_and_Chronos?=
In-Reply-To: <1183309519.4687decf17576@imp.free.fr>
References: <1183283557.468779659d8ca@imp.free.fr>
	 <5542af940707010928h4890e4b4ofa519b20a082694@mail.gmail.com>
	 <1183309519.4687decf17576@imp.free.fr>
Message-ID: <5542af940707011018g6dff5435w42ece630acbb0496@mail.gmail.com>

I speak of your mathematical work: "It appears self consistent,
correct, and, to me, interesting."

Regarding Time, I have taught several hundred adult students a course
on "Time Travel: Math, Physics, Fiction" using as textbook:

Time Machines: Time Travel in Physics, Metaphysics, and Science
Fiction, by Paul J. Nahin, Springer-Verlag New York, 1993.

There is greater clarity in studing primes, and asymptotic limits of
real functions, than in the Philosophy of Time, or, perhaps, any
Philosophy.  So I shall for some time restrict my seqfans comments to
Mathematics and Integers.

Classically:

"... But the position of these and similar authorities is made clear
by Boethius, who says (V De Consolatione prosa 6), "When some people
hear that Plato thought this world neither had a beginning in time nor
will ever have an end, they mistakenly conclude that the created world
is coeternal with the Creator. However, to be led through the endless
life Plato attributes to the world is one thing; to embrace
simultaneously the whole presence of endless life is quite another,
and it is this latter that is proper to the divine mind." [PL 63,
859B]

<a href="http://www.fordham.edu/halsall/basis/aquinas-eternity.html">Medieval
Sourcebook:
Thomas Aquinas:
On The Eternity of the World
(DE AETERNITATE MUNDI)
DE AETERNITATE MUNDI [[1]]
Translation (c) 1991, 1997 by Robert T. Miller[[2]] </a>



Since people are posting sequence puzzles on seq.fan lately, I thought I 
would post this sequence puzzle of a different varity.

I suspect this 'puzzle' is easy, and I'll probably regret I posted this.

---

Let {c(k)} be as defined at sequence A022940. ({c(k)} itself is not in 
the EIS.)

Define sequence {a(k)} as follows:

Let b(n) = c(n) - n + 1.

a(1) = the number of 1's in {b(k)}. a(2) = the number of 2's in {b(k)}.
In general, a(n) = the number of n's in {b(k)}.

So, {a(k)} begins: 0,1,1,3,5,6,7,9,...

Define {a(k)}. (Define it in a simpler way than by the steps given above.)

Thanks,
Leroy Quet




From qq-quet at mindspring.com  Mon Jul  2 00:42:07 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Sun, 1 Jul 07 16:42:07 -0600
Subject: Another kind of sequence puzzle
References: <1183283557.468779659d8ca@imp.free.fr>
Message-ID: <E1I58AC-0006iC-00@pop03.mail.atl.earthlink.net>

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

SeqFiends,

I can't recall the context, but we have several times over
the past few years been discussing several questions about
boolean functions B^k -> B and boolean mappings B^k -> B^n.

I am finally getting some of the work that I referred to
in slightly prettier shape.  Here's a link to a paper on
Differential Logic:

http://www.centiare.com/Differential_Logic_and_Dynamic_Systems

Related material can be found by perusing this directory page:

http://www.centiare.com/Directory:Jon_Awbrey

Some of this work even has a little bit to do with the
love that Eternity bears toward the creatures of Time.

Many Regards,

Jon Awbrey

CC: Arisbe Forum: http://stderr.org/pipermail/arisbe/
CC: Inquiry List: http://stderr.org/pipermail/inquiry/

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
inquiry e-lab: http://stderr.org/pipermail/inquiry/
?iare: http://www.centiare.com/Directory:Jon_Awbrey
getwiki: http://www.getwiki.net/-User_talk:Jon_Awbrey
zhongwen wp: http://zh.wikipedia.org/wiki/User:Jon_Awbrey
http://www.altheim.com/ceryle/wiki/Wiki.jsp?page=JonAwbrey
wp review: http://wikipediareview.com/index.php?showuser=398
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o





From Eric.Angelini at kntv.be  Mon Jul  2 17:28:54 2007
From: Eric.Angelini at kntv.be (Eric Angelini)
Date: Mon, 2 Jul 2007 17:28:54 +0200
Subject: Seq and first diff show the same "digit pattern"
Message-ID: <F05775148D000C4B9321A5113D53CB7D405D2B@KNTV-SERVER01.kntv.local>


Hello SeqFan,
could someone compute a few more terms of this seq:

1  12  14  155  160  211  271  292  419  548  572  691 ...

The principle is:

- Seq and first diff show the same "digit pattern".

S = 1  12  14   155  160  211  271  292   419   548  572  691 ...
d =  11   2  141    5   51   60   21   127   129   24   19 ...

Rules:
- start S with "1"
- add to the last term of S the smallest integer d no yet
  added and not present in S such that the concatenation
  of S's terms and the concatenation of all ds are the
  same string of digits

So, never twice the same integer in sequence or first
differences. 

I'm quite sure that all N's will be split between S and d.

Best,
?.

http://www.research.att.com/~njas/sequences/A110621  
has a close Mathematica pgm by Robert G. Wilson.

(thanks again to him!)




Something is wrong with my email, and I haven't been receiving all emails 
have.

I will post the solution now, even though only a short while has passed 

Original email as spoiler-space.

>Since people are posting sequence puzzles on seq.fan lately, I thought I 
>would post this sequence puzzle of a different varity.
>
>I suspect this 'puzzle' is easy, and I'll probably regret I posted this.
>
>---
>
>Let {c(k)} be as defined at sequence A022940. ({c(k)} itself is not in 
>the EIS.)
>
>Define sequence {a(k)} as follows:
>
>Let b(n) = c(n) - n + 1.
>
>a(1) = the number of 1's in {b(k)}. a(2) = the number of 2's in {b(k)}.
>In general, a(n) = the number of n's in {b(k)}.
>
>So, {a(k)} begins: 0,1,1,3,5,6,7,9,...
>
>Define {a(k)}. (Define it in a simpler way than by the steps given above.)
>
>Thanks,
>Leroy Quet


I get that a(1)=0, a(2)=1. a(n) = c(n-2) -1, for n >= 3.

I don't know if there is a "closed form" (nonrecursive representation) 
for {c(k)}.

Thanks,
leroy Quet




From qq-quet at mindspring.com  Mon Jul  2 19:29:58 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Mon, 2 Jul 07 11:29:58 -0600
Subject: Another kind of sequence puzzle
Message-ID: <E1I5Ple-0005xt-00@pop06.mail.atl.earthlink.net>

sent to me. So I don't know if someone has posted the solution before I 
since I posted the question.
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Message-ID: <fe71ae2b0707021836s19391bdaqd911c68dd9b3f373 at mail.gmail.com>
Date: Mon, 2 Jul 2007 20:36:10 -0500
From: xordan <xordan.tom at gmail.com>
To: seqfan at ext.jussieu.fr
Subject: relatively primes and prime numbers graphs.
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I understand that the attached file and the folloging words that  don't
coincide with the strict discussion line in seqfan, but they contain some
graphic curiosities that I wanted to share with the members of the list. I
wait some benevolent comments. Again I notice that the translation of the
original in Spanish is made with the help of software:

The attached file (coprimes.zip ) contains one book of calculation sheets
that has 5 work sheets that give results (to my view) interesting related
with the numbers relatively primes.
The first sheet shows the numbers (1 to 256) relatively primes to each
other  that added (arithmetic sum)   becomes  a prime number as  result; the
second sheet shows the numbers relatively primes  whose absolute difference
becomes a prime number  as  result; the third are the conjunction of the
previous two , that is to say the numbers relatively primes  whose their
algebraic sum  gives as  result a prime  number. The fourth is the same
graph that it appears in the current page
http://mathworld.wolfram.com/RelativelyPrime.html (RelativelyPrime.gif) and
that it shows the primes numbers relatively to each other. The fifth work
sheet is the conjunction of RelativelyPrime.gif and the previous sheet
"abs(r+-c)=prime". - With this it is shown graphically that all the numbers
(r,c) whose algebraic sum is a prime number (p) they are relatively prime to
each other.
IF  r+c=p  THEN   coprime(r,c)=1.-
-- 
xordan at hotmail.com
xordan_co at yahoo.com
xordan.tom at gmail.com

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<div>I understand that the attached file and the folloging words that&nbsp; don&#39;t coincide with the strict discussion line in seqfan, but they contain some graphic curiosities that&nbsp;I wanted to share with the members of the list. I wait some benevolent comments. Again I notice that the translation of the original in Spanish is made with the help of software: 
</div>
<div>&nbsp;</div>
<div>The attached file (coprimes.zip )&nbsp;contains&nbsp;one book of calculation sheets that has 5 work&nbsp;sheets that give results (to my view) interesting related with the numbers relatively primes.&nbsp;&nbsp; <br>The first sheet shows the numbers (1&nbsp;to 256) relatively primes to each other&nbsp; that added (arithmetic sum)&nbsp;&nbsp; becomes&nbsp; a prime number as&nbsp; result; the second sheet shows the numbers relatively primes&nbsp; whose absolute difference becomes a prime number&nbsp; as&nbsp; result; the third are the&nbsp;conjunction of the previous two , that is to say the numbers relatively primes&nbsp; whose their&nbsp; algebraic sum&nbsp; gives as&nbsp; result a prime&nbsp; number. The fourth is the same graph that it appears in the current page 
<a href="http://mathworld.wolfram.com/RelativelyPrime.html">http://mathworld.wolfram.com/RelativelyPrime.html</a>&nbsp;(RelativelyPrime.gif) and that it shows the primes numbers relatively to each other. The fifth work sheet is the conjunction of 
RelativelyPrime.gif and the previous sheet &quot;abs(r+-c)=prime&quot;. - With this it is shown graphically that all the numbers (r,c) whose algebraic sum is a prime number (p) they are relatively prime to each other.</div>

<div>IF&nbsp;&nbsp;r+c=p&nbsp;&nbsp;THEN&nbsp;&nbsp;&nbsp;coprime(r,c)=1.- <br>-- <br><a href="mailto:xordan at hotmail.com">xordan at hotmail.com</a><br><a href="mailto:xordan_co at yahoo.com">xordan_co at yahoo.com</a><br><a href="mailto:xordan.tom at gmail.com">xordan.tom at gmail.com
</a> </div>

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------=_Part_104140_14761177.1183426570634--



Maximilian Hasler has pointed out to me that at sequence A022940 
"complement" is not defined.

I assume that Clark Kimberling intended {c(k)} to be those positive 
integers (in order) that do not appear in A022940.

So {c(k)} begins: 2,4,6,7,8,10,11,12,13,14,16,17,...

Thanks,
Leroy Quet



Hello Seqfans,

I ordered 2 sequence numbers. I didn't receive the confirmation email and on the page with numbers there is the following paragraph:

Insufficient disk space; try again later Insufficient disk space; try again later returntosender: cannot select queue for oeis Insufficient disk space; try again later returntosender: cannot select queue for postmaster putbody: write error: No space left on device Error writing control file qfl63ErU4Y019338: No space left on device 

Is OEIS working?
Can I use my numbers?

Tanya


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From qq-quet at mindspring.com  Tue Jul  3 17:09:05 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Tue, 3 Jul 07 09:09:05 -0600
Subject: Another kind of sequence puzzle
In-Reply-To: <721e81490707022117o386e3adau95c08683f9d9aa35@mail.gmail.com>
References: <fe71ae2b0707021836s19391bdaqd911c68dd9b3f373@mail.gmail.com>
	 <721e81490707022117o386e3adau95c08683f9d9aa35@mail.gmail.com>
Message-ID: <E1I5k2r-0006Ep-00@pop04.mail.atl.earthlink.net>

------=_Part_114770_16554780.1183484165194
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Content-Transfer-Encoding: 7bit
Content-Disposition: inline

Yes Joshua:
The only exception ocurrs if r | c or if c | r in the case abs(r-c)=prime;
in other words if r and c are multiples of p.


2007/7/2, Joshua Zucker <joshua.zucker at gmail.com>:
>
> Proof: if r and c are not coprime, then they have a common divisor d,
> and then d divides r+c and r-c as well.
>
> The only possible exceptions would be cases like 6 and 3, where 6-3 is
> prime even though 6 and 3 are not coprime.  That is, where the common
> divisor is a prime equal to r-c.
>
> --Joshua Zucker
>
>
> On 7/2/07, xordan <xordan.tom at gmail.com> wrote:
> > I understand that the attached file and the folloging words that  don't
> > coincide with the strict discussion line in seqfan, but they contain
> some
> > graphic curiosities that I wanted to share with the members of the list.
> I
> > wait some benevolent comments. Again I notice that the translation of
> the
> > original in Spanish is made with the help of software:
> >
> > The attached file (coprimes.zip ) contains one book of calculation
> sheets
> > that has 5 work sheets that give results (to my view) interesting
> related
> > with the numbers relatively primes.
> > The first sheet shows the numbers (1 to 256) relatively primes to each
> other
> >  that added (arithmetic sum)   becomes  a prime number as  result; the
> > second sheet shows the numbers relatively primes  whose absolute
> difference
> > becomes a prime number  as  result; the third are the conjunction of the
> > previous two , that is to say the numbers relatively primes  whose their
> > algebraic sum  gives as  result a prime  number. The fourth is the same
> > graph that it appears in the current page
> > http://mathworld.wolfram.com/RelativelyPrime.html
> > (RelativelyPrime.gif) and that it shows the primes numbers relatively to
> > each other. The fifth work sheet is the conjunction of
> RelativelyPrime.gif
> > and the previous sheet "abs(r+-c)=prime". - With this it is shown
> > graphically that all the numbers (r,c) whose algebraic sum is a prime
> number
> > (p) they are relatively prime to each other.
> > IF  r+c=p  THEN   coprime(r,c)=1.-
> > --
> > xordan at hotmail.com
> > xordan_co at yahoo.com
> > xordan.tom at gmail.com
> >
>



-- 
xordan at hotmail.com
xordan_co at yahoo.com
xordan.tom at gmail.com

------=_Part_114770_16554780.1183484165194
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<div>Yes Joshua:</div>
<div>The only exception ocurrs if r | c&nbsp;or if c | r in the case abs(r-c)=prime; in other words if r and c&nbsp;are multiples of p.&nbsp;</div>
<div>&nbsp;<br>&nbsp;</div>
<div><span class="gmail_quote">2007/7/2, Joshua Zucker &lt;<a href="mailto:joshua.zucker at gmail.com">joshua.zucker at gmail.com</a>&gt;:</span>
<blockquote class="gmail_quote" style="PADDING-LEFT: 1ex; MARGIN: 0px 0px 0px 0.8ex; BORDER-LEFT: #ccc 1px solid">Proof: if r and c are not coprime, then they have a common divisor d,<br>and then d divides r+c and r-c as well.
<br><br>The only possible exceptions would be cases like 6 and 3, where 6-3 is<br>prime even though 6 and 3 are not coprime.&nbsp;&nbsp;That is, where the common<br>divisor is a prime equal to r-c.<br><br>--Joshua Zucker<br><br><br>
On 7/2/07, xordan &lt;<a href="mailto:xordan.tom at gmail.com">xordan.tom at gmail.com</a>&gt; wrote:<br>&gt; I understand that the attached file and the folloging words that&nbsp;&nbsp;don&#39;t<br>&gt; coincide with the strict discussion line in seqfan, but they contain some
<br>&gt; graphic curiosities that I wanted to share with the members of the list. I<br>&gt; wait some benevolent comments. Again I notice that the translation of the<br>&gt; original in Spanish is made with the help of software:
<br>&gt;<br>&gt; The attached file (coprimes.zip ) contains one book of calculation sheets<br>&gt; that has 5 work sheets that give results (to my view) interesting related<br>&gt; with the numbers relatively primes.<br>&gt; The first sheet shows the numbers (1 to 256) relatively primes to each other
<br>&gt;&nbsp;&nbsp;that added (arithmetic sum)&nbsp;&nbsp; becomes&nbsp;&nbsp;a prime number as&nbsp;&nbsp;result; the<br>&gt; second sheet shows the numbers relatively primes&nbsp;&nbsp;whose absolute difference<br>&gt; becomes a prime number&nbsp;&nbsp;as&nbsp;&nbsp;result; the third are the conjunction of the
<br>&gt; previous two , that is to say the numbers relatively primes&nbsp;&nbsp;whose their<br>&gt; algebraic sum&nbsp;&nbsp;gives as&nbsp;&nbsp;result a prime&nbsp;&nbsp;number. The fourth is the same<br>&gt; graph that it appears in the current page<br>&gt; <a href="http://mathworld.wolfram.com/RelativelyPrime.html">
http://mathworld.wolfram.com/RelativelyPrime.html</a><br>&gt; (RelativelyPrime.gif) and that it shows the primes numbers relatively to<br>&gt; each other. The fifth work sheet is the conjunction of RelativelyPrime.gif<br>
&gt; and the previous sheet &quot;abs(r+-c)=prime&quot;. - With this it is shown<br>&gt; graphically that all the numbers (r,c) whose algebraic sum is a prime number<br>&gt; (p) they are relatively prime to each other.<br>
&gt; IF&nbsp;&nbsp;r+c=p&nbsp;&nbsp;THEN&nbsp;&nbsp; coprime(r,c)=1.-<br>&gt; --<br>&gt; <a href="mailto:xordan at hotmail.com">xordan at hotmail.com</a><br>&gt; <a href="mailto:xordan_co at yahoo.com">xordan_co at yahoo.com</a><br>&gt; <a href="mailto:xordan.tom at gmail.com">
xordan.tom at gmail.com</a><br>&gt;<br></blockquote></div><br><br clear="all"><br>-- <br><a href="mailto:xordan at hotmail.com">xordan at hotmail.com</a><br><a href="mailto:xordan_co at yahoo.com">xordan_co at yahoo.com</a><br><a href="mailto:xordan.tom at gmail.com">
xordan.tom at gmail.com</a> 

------=_Part_114770_16554780.1183484165194--



Are other people also having problems receiving the automated reply when 
with this.)

I guess we will have to wait until Neil gets back from vacation to submit 
via this method.

Thanks,
Leroy Quet






From qq-quet at mindspring.com  Wed Jul  4 18:16:42 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Wed, 4 Jul 07 10:16:42 -0600
Subject: No automatic replies
Message-ID: <E1I67Zp-0002UH-00@pop05.mail.atl.earthlink.net>

submitting sequences via the on-line form? (I know Tanya has had problems 
Return-Path: <jvospost3 at gmail.com>
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Message-ID: <5542af940707041259v435e0f9alf913531e86f7230c at mail.gmail.com>
Date: Wed, 4 Jul 2007 12:59:45 -0700
From: "Jonathan Post" <jvospost3 at gmail.com>
To: "Leroy Quet" <qq-quet at mindspring.com>,
   "jonathan post" <jvospost2 at yahoo.com>
Subject: Re: No automatic replies
Cc: seqfan at ext.jussieu.fr
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X-Miltered: at shiva.jussieu.fr with ID 468BFC32.001 by Joe's j-chkmail (http://j-chkmail.ensmp.fr)!

I got no autoreply for this:

NEW SEQUENCE FROM Jonathan Vos Post

%I A000001
%S A000001 -109, -42, -26, -18, -14, -12, -10, -9, -8,
-7, -6, -6, -5, -5, -5, -4, -4, -4, -4, -4, -3, -3,
-3, -3, -3, -3, -3, -3, -2, -2, -2, -2, -2, -2, -2,
-2, -2, -2, -2, -2, -2, -2, -2, -1
%N A000001 Floor of constants in De Bruijn's approach
to weighted Carleman's inequality.
%C A000001 Abstract of Gao: "We study finite sections
of weighted Carleman's inequality following the
approach of De Bruijn. Similar to the unweighted case,
we obtain an asymptotic expression for the optimal
constant."
%D A000001 N. G. De Bruijn, Carleman's inequality for
finite series, Nederl. Akad. Wetensch. Proc. Ser. A 66
= Indag, pp. 505-514.
%H A000001 Peng Gao, <a
href="http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0077v1.pdf">Finite
Sections of Weighted Carleman's Inequality</a>, 30
June 2007, p.1.
%F A000001 a(n) = floor(e -
(2*(pi^2)*e)/((log(n))^2)).
%e A000001 a(2) = -109 because e -
(2*(pi^2)*e)/((log(2))^2) ~
-108.9611770171388392925257212314455433803548032218666994709.
a(3) = -42 because e - (2*(pi^2)*e)/((log(3))^2) ~
-41.7382232411477828847325690963577817095329948893743754723.
a(4) = -26 because e - (2*(pi^2)*e)/((log(4))^2) ~
-25.20158288294042589661121470434688897177076548519170518650.
a(30) = -2 because e - (2*(pi^2)*e)/((log(30))^2) ~
-1.92003649778404604739381818236913112747520.
a(45) = -1 because e - (2*(pi^2)*e)/((log(45))^2) ~
-0.98456269963010489451493724472555817336322761419762175593.
%O A000001 2,1
%K A000001 ,easy,sign,
%A A000001 Jonathan Vos Post (jvospost2 at yahoo.com),
Jul 03 2007




From jvospost3 at gmail.com  Wed Jul  4 22:01:04 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Wed, 4 Jul 2007 13:01:04 -0700
Subject: No automatic replies
In-Reply-To: <5542af940707041259v435e0f9alf913531e86f7230c@mail.gmail.com>
References: <E1I67Zp-0002UH-00@pop05.mail.atl.earthlink.net>
	 <5542af940707041259v435e0f9alf913531e86f7230c@mail.gmail.com>
Message-ID: <5542af940707041301s32f82ca9h2f1e3638a33ddecc@mail.gmail.com>

Nor did I get an autoreply for this:

COMMENT FROM Jonathan Vos Post RE A122505

%I A122505
%S A122505 24, 24, 95, 1, 143, 1, 262, -213, 453,
-261, 739, -833, 1169, -1168, 2172, -2505, 3104,
-3581, 5255, -6449
%N A122505 Arises from energy spectrum of three
dimensional gravity with negative cosmological
constant, in analysis by Edward Witten.
%C A122505 a(11)-a(20) given by "GH" at Not Even Wrong
Blog; not verified as correct; commenter email
unknown.
%H A122505 "GH", <a
href="http://www.math.columbia.edu/~woit/wordpress/?p=571#comment-26500">Strings
2007</a>, thread of "Not Even Wrong" Physics blog,
edited by Peter Woit, <woit at math.columbia.edu>, thread
initiated 25 June 2007, comment by "GH" extending
sequence dated 1 July 2007.
%O A122505 1
%K A122505 ,sign,
%A A122505 Jonathan Vos Post (jvospost2 at yahoo.com),
Jul 03 2007




From jvospost3 at gmail.com  Wed Jul  4 22:05:27 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Wed, 4 Jul 2007 13:05:27 -0700
Subject: No automatic replies
In-Reply-To: <5542af940707041301s32f82ca9h2f1e3638a33ddecc@mail.gmail.com>
References: <E1I67Zp-0002UH-00@pop05.mail.atl.earthlink.net>
	 <5542af940707041259v435e0f9alf913531e86f7230c@mail.gmail.com>
	 <5542af940707041301s32f82ca9h2f1e3638a33ddecc@mail.gmail.com>
Message-ID: <5542af940707041305r34b54aa2w219a86bc732a103a@mail.gmail.com>

For that matter (and I'm not saying that ANY of these are good or
important, just that they track to OEIS and the Literature):

COMMENT FROM Jonathan Vos Post RE A001399

%I A001399
%S A001399 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30,
33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108,
114, 120, 127, 133, 140, 147, 154, 161, 169, 176, 184, 192, 200, 208,
217, 225, 234, 243, 252, 261, 271, 280, 290, 300, 310, 320, 331, 341
%N A001399 Number of partitions of n into at most 3 parts; also
partitions of n+3 in which the greatest part is 3; also multigraphs
with 3 nodes and n edges.
%H A001399 Andrew N. Norris, <a
href="http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0115v1.pdf">
Higher derivatives and the inverse derivative of a tensor-valued
function of a tensor</a>, 1 July 2007, Equation 3.28, p.10
%F A001399 After initial 1 appears identical to integer part of
((n+4)^2 + 4)/12, which is given Norris as the number of points in,
and on the boundary of the integer grid of {I, J}, bounded by the
three straight lines I = 0, I - J = 0, and I + 2J = n + 1.
%O A001399 1
%K A001399 ,nonn,
%A A001399 Jonathan Vos Post (jvospost2 at yahoo.com), Jul 03 2007




From martin_n_fuller at btinternet.com  Fri Jul  6 21:34:13 2007
From: martin_n_fuller at btinternet.com (Martin Fuller)
Date: Fri, 6 Jul 2007 20:34:13 +0100 (BST)
Subject: Help correcting A003294 and A096739
Message-ID: <324718.39838.qm@web86601.mail.ird.yahoo.com>

SeqFans,

A003294 and A096739 have some obvious errors, but they may also be the same
sequence.  Can anyone help provide terms and/or decide whether they are
duplicates?

http://www.research.att.com/~njas/sequences/?q=id:A003294|id:A096739

The first error is that there are missing multiples, e.g. 1765 = 353*5.  Then
there is a typo in A096739 which has 5129 instead of 5729.  It looks like the
values have been brought together from several sources but not checked
carefully enough.

Are there any fourth power sums which involve repeated values?  If the answer
is proved to be no, then these sequences can be combined.

Martin Fuller




From maximilian.hasler at gmail.com  Sat Jul  7 00:33:01 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Fri, 6 Jul 2007 18:33:01 -0400
Subject: Help correcting A003294 and A096739
In-Reply-To: <324718.39838.qm@web86601.mail.ird.yahoo.com>
References: <324718.39838.qm@web86601.mail.ird.yahoo.com>
Message-ID: <3c3af2330707061533m3328ccb0vec62ccd85c7db644@mail.gmail.com>

> http://www.research.att.com/~njas/sequences/?q=id:A003294|id:A096739
>
> The first error is that there are missing multiples, e.g. 1765 = 353*5.

I think this is an omission in the definition rather than error in the
sequence -
I believe in general for such "homogeneous" diophantine equations
it is somehow tacitly understood that multiples are not listed as solutions
(since any multiple of a solution (A,B,C,D,n) is trivially again a solution,
one only gives solutions with gcd(A,B,C,D,n) = 1.)

but of course this should be somehow said in the definition.

M.H.




From jvospost3 at gmail.com  Sat Jul  7 00:33:48 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Fri, 6 Jul 2007 15:33:48 -0700
Subject: Correction to, comment on, A061789, and primes therein
Message-ID: <5542af940707061533l7a0654a7u2ef96454c2cae8db@mail.gmail.com>

Correction to A061789:

a(4) = 2^2 + 3^3 + 5^5 + 7^7 = 826699
rather than, as shown, 826696.

Comment:

Primes in A061689

a(2) = 2^2 + 3^3 = 31 is prime

a(4) = 2^2 + 3^3 + 5^5 + 7^7 = 826699 is prime

a(24) = 2^2 + 3^3 + 5^5 + 7^7 + 11^11 + 13^13 + 17^17 + 19^19
+ 23^23 + 29^29 + 31^31 + 37^37 + 41^41 + 43^43 +
47^47 + 53^53 + 59^59 + 61^61 + 67^67 + 71^71 + 73^73
+ 79^79 + 83^83 + 89^89 =
313119 843606 266222 723931 946804 611259 287100
533754 507329 247768 328563 715123 287679 948437
264517 293404 303260 795541 448480 275059 086587
161357 892405 477908 284669 463639 709544 036228
659629 is prime

A061789 Sum_{k=1..n} p(k)^p(k), p(k) = k-th prime (A000040).
       4, 31, 3156, 826696, 285312497310, 303160419089563,
827240565046755853740, 1979246896225360344977719,
20880469979094808259715377888286,
2567686153182091604540923022990731504371755




From djr at nk.ca  Sat Jul  7 04:46:18 2007
From: djr at nk.ca (Don Reble)
Date: Fri, 06 Jul 2007 20:46:18 -0600
Subject: Help correcting A003294 and A096739
In-Reply-To: <324718.39838.qm@web86601.mail.ird.yahoo.com>
References: <324718.39838.qm@web86601.mail.ird.yahoo.com>
Message-ID: <468EFE7A.1040400@nk.ca>

Seqfans:

    A003294 goes
         353  651  706 1059 1302 1412 1765 1953 2118 2471
        2487 2501 2604 2824 2829 3177 3255 3530 3723 3883
        3906 3973 4236 4267 4333 4449 4557 4589 4942 4949
        4974 5002 5208 5281 5295 5463 5491 5543 5648 5658
        5729 5859 6001 6167 6354 6510 6609 6707 6801 7060
        7101 7161 7209 7339 7413 7446 7461 7503 7703 7766
        7812 7946 8119 8373 8433 8463 8472 8487 8493 8517
        8534 8577 8637 8666 8825 8898 9114 9137 9178 9243
        9431 9519 9531 9639 9765 9797 9877 9884 9898 9948
    and all of these are in A096739. So they still might be the same.

    One might make a sequence of the primitive terms, where the fourth
    roots have no common factor. Ah! that's the intent of A039664.
    Dr. Sloane, do you have the original author?

    I'll send A003294 and A039664 edits to Dr. Sloane.

-- 
Don Reble  djr at nk.ca



-- 
This message has been scanned for viruses and
dangerous content by MailScanner, and is
believed to be clean.





From jvospost3 at gmail.com  Mon Jul  9 00:31:49 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Sun, 8 Jul 2007 15:31:49 -0700
Subject: Correction to, comment on, A061789, and primes therein
In-Reply-To: <5542af940707061533l7a0654a7u2ef96454c2cae8db@mail.gmail.com>
References: <5542af940707061533l7a0654a7u2ef96454c2cae8db@mail.gmail.com>
Message-ID: <5542af940707081531o25ddaca4g5019982b3f02d3d@mail.gmail.com>

I've verified, using Alpertron, and double-checking with the WIMS
server at U.Nice, that there are no more primes in A061789 through
a(78):

2^2 + 3^3 + 5^5 + 7^7 + ... + 397^397

which has 1032 digits.

hence, if there is a 4th prime in A061789 it must be a Titanic prime,
to use the term coined by Samuel Yates in the 1980s, denoting a prime
number of at least 1000 decimal digits.




From jvospost3 at gmail.com  Tue Jul 10 02:03:30 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Mon, 9 Jul 2007 17:03:30 -0700
Subject: Prime Fermat number-twins (Boris V. Tarasov, arXiv)
Message-ID: <5542af940707091703r69cc1411h385ccd4a7919158e@mail.gmail.com>

Is it worth submitting this sequence?

Prime Fermat number-twins.

5, 7, 13, 19, 65539

Prime numbers of the form (2^(2^n)) - 3 or (2^(2^n)) + 3

The concrete theory of numbers : Problem of simplicity of Fermat number-twins
http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0907v1.pdf
Authors: Boris V. Tarasov
Comments: 6 pages
Subjects: General Mathematics (math.GM)

The problem of simplicity of Fermat number-twins f_n^{plus or minus) =
2^(2^n){plus or minus 3} is studied. The question for what n numbers
f_n^{plus or minus) are composite is investigated. The
factor-identities for numbers of a kind x^2 {plus or minus} k $ are
found.

a(n) are (sorted) from Tarasov, p.5

If this is worth submitting, what is the next value a(5) asserted by
Tarasov to be equal or greater than (2^(2^17)) - 3?




From jvospost3 at gmail.com  Tue Jul 10 02:10:47 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Mon, 9 Jul 2007 17:10:47 -0700
Subject: Marek Wolf, arXiv: Analog of the Skewes number for twin primes
Message-ID: <5542af940707091710o27abe8ft727a16b924a95413@mail.gmail.com>

Dear Marek Wolf,

I certainly hope that you submit sequence(s) to the Online
Encyclopedia of IUnteger Sequences from your fine paper:

arXiv:0707.0980
    Title: Analog of the Skewes number for twin primes
    Authors: Marek Wolf
    Subjects: Number Theory (math.NT)
6 July 2007

Sincerely,

Prof. Jonathan Vos Post




From zakseidov at yahoo.com  Tue Jul 10 15:57:55 2007
From: zakseidov at yahoo.com (zak seidov)
Date: Tue, 10 Jul 2007 06:57:55 -0700 (PDT)
Subject: Concise Search Result
In-Reply-To: <5542af940707041305r34b54aa2w219a86bc732a103a@mail.gmail.com>
Message-ID: <93596.85791.qm@web38208.mail.mud.yahoo.com>

Dear seqfans,

I think that Concise Search Result
of the following form would be nice, right?
Zak
 
Search: 18, 96, 600, 4320
Result:
A001563, A052633,  A052655, A094258, A094304, A109074


       
____________________________________________________________________________________
Need a vacation? Get great deals
to amazing places on Yahoo! Travel.
http://travel.yahoo.com/




From ogerard at ext.jussieu.fr  Tue Jul 10 18:42:39 2007
From: ogerard at ext.jussieu.fr (=?ISO-8859-1?Q?Olivier_G=E9rard?=)
Date: Tue, 10 Jul 2007 18:42:39 +0200
Subject: [seqfan] INTERRUPTION DE SERVICE - MAILING LIST HIATUS
Message-ID: <4693B6FF.3010103@ext.jussieu.fr>



Dear list members,


The machine hosting the seqfan list server will be shut down from

Thursday July 12 6pm GMT+2    until    Friday July 13 9am GMT+2

(that's the end of tuesday morning till 1 or 2am friday US East time)



Please do not send any message during this period as you will
not be able to check whether it has been received or not by
the mailing list robot.  Anyway, they would be distributed
only after the servers are rebooted.

I will send a mail to the list so that everyone knows operations
are back to normal.  Thanks in advance for your patience and
your discipline in that matter.

with my best regards,

Olivier GERARD

PS: As usual, any inquiries about this matter should be directed
to me (olivier.gerard at gmail.com or olivier.gerard at paris7.jussieu.fr)
and not to the list.

===============================
===============================

Chers abonn?s de la liste seqfan

Arr?t temporaire du serveur h?bergeant la liste

JEUDI 12 Juillet 18h, heure de Paris, pour les fran?ais,

jusqu'au

VENDREDI 13 Juillet au matin (probablement 9h)


S'il vous plait, n'envoyez pas de message pendant cette p?riode,
vous ne pourriez pas savoir si il a ?t? bien re?u, et vous risqueriez
de provoquer une confusion ou des doublons.
J'enverrai un courier ? la liste quand la situation sera revenue
? la normale.

Merci d'avance pour votre patience et votre discipline.


Cordialement,

Olivier GERARD


PS: Comme d'habitude, si vous avez des questions ? ce sujet, ?crivez
moi directement (olivier.gerard at gmail.com ou 
olivier.gerard at paris7.jussieu.fr ) et pas ? la liste.







From diana.mecum at gmail.com  Wed Jul 11 15:22:53 2007
From: diana.mecum at gmail.com (Diana Mecum)
Date: Wed, 11 Jul 2007 08:22:53 -0500
Subject: Question related to sequence A071267
Message-ID: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>

Sequence Fans,

I am looking at sequence A071267.

%I A071267
%S A071267
2,4,6,8,10,11,12,14,16,18,22,33,44,55,66,77,88,99,101,110,121,132,143,
%T A071267 154,165,176,187,198,111,222,666,888,1110
%N A071267 Numbers which can be expressed as the sum of all distinct digit
permutations of
               some number k.
%C A071267 222 can be expressed so in two different ways i.e. 222= 200 +020
+ 002 as well as
               222= 101 +110 +011. Question: find a number which can be so
expressed
               in n different ways.
%e A071267 1110 is a member as a sum of all distinct permutations of 104.
i.e. 104,140,410,
               401,014,041.
%Y A071267 Sequence in context: A081472 A097660 A067030 this_sequence
A072427 A050420 A096922
%Y A071267 Adjacent sequences: A071264 A071265 A071266 this_sequence A071268
A071269 A071270
%K A071267 base,more,nonn
%O A071267 1,1
%A A071267 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 01 2002

When I try to follow the rule for generating the sequence numbers, I get the
following list;

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
121, 132, 143, 154, 165,
 176, 187, 222, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443, 1554,
1665, 1776, 1887,
 1998, 2109, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996

Can someone explain why my list differs from the original? I am not
understanding the hypothesis to generate the original list.

Diana M.

-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From joshua.zucker at gmail.com  Wed Jul 11 17:12:36 2007
From: joshua.zucker at gmail.com (Joshua Zucker)
Date: Wed, 11 Jul 2007 08:12:36 -0700
Subject: Question related to sequence A071267
In-Reply-To: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>
References: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>
Message-ID: <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com>

Hi Diana and all,
Murthy's sequences often have errors.  Here, among other problems, the
definition is "numbers which ..." but then shouldn't they be SORTED?

I don't quite agree with your list, either, though.  Here's what I
think, based purely on brute-force with a computer, using as "seeds"
all the numbers up to 100000:

1 2 3 4 5 6 7 8 9 11 22 33 44 55 66 77 88 99 110 111 121 132 143 154
165 176 187 222 333 444 555 666 777 888 999 1110 1111 1221 1332 1443
1554 1665 1776 1887 1998 2109 2220 2222 2331 2442 2553 2664 2775 2886
3108 3330 3333 3552 3774 3996 4218 4440 4444 4662 4884 5106 5328 5555
6666 7777 8888 9999 11110 11111 12221 13332 14443 15554 16665 17776
18887 19998 21109 22220 22222 23331 24442 25553 26664 27775 28886
29997

which leaves me wondering, for instance, why I got 29997 but not 2997
in there.  But now I see it's a nice little bit of combinatorics: 108
goes to 1998 (six permutations, so effectively each spot is
(1+0+8+1+0+8) so we get 1800 + 180 + 18), while 1008 goes to 29997
(twelve permutations, so each spot is (1+0+0+8+1+0+0+8+1+0+0+8) so we
get 27000 + 2700 + 270 + 27).

So I think my above list of terms are correct.  Differences from your
list, Diana, are that I have 333 and 1111 and 2222 and 2331 and 2553
and 2775 and 3333 which are missing from your list.  (2331 for example
comes from 399 -> 399 + 939 + 993, and any list of identical digits
maps to itself, so 333 -> 333.)

Diana, would you verify that you agree with my list of terms and then
if you do please submit the corrected terms for this sequence?

Thanks,
--Joshua Zucker


On 7/11/07, Diana Mecum <diana.mecum at gmail.com> wrote:
> Sequence Fans,
>
> I am looking at sequence A071267.
>
> %I A071267
> %S A071267
> 2,4,6,8,10,11,12,14,16,18,22,33,44,55,66,77,88,99,101,110,121,132,143,
> %T A071267 154,165,176,187,198,111,222,666,888,1110
> %N A071267 Numbers which can be expressed as the sum of all distinct digit
> permutations of
>                some number k.
> %C A071267 222 can be expressed so in two different ways i.e. 222= 200 +020
> + 002 as well as
>                222= 101 +110 +011. Question: find a number which can be so
> expressed
>                 in n different ways.
> %e A071267 1110 is a member as a sum of all distinct permutations of 104.
> i.e. 104,140,410,
>                401,014,041.
> %Y A071267 Sequence in context: A081472 A097660 A067030 this_sequence
> A072427 A050420 A096922
> %Y A071267 Adjacent sequences: A071264 A071265 A071266 this_sequence A071268
> A071269 A071270
> %K A071267 base,more,nonn
> %O A071267 1,1
> %A A071267 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 01 2002
>
> When I try to follow the rule for generating the sequence numbers, I get the
> following list;
>
> 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
> 121, 132, 143, 154, 165,
>  176, 187, 222, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443, 1554,
> 1665, 1776, 1887,
>  1998, 2109, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996
>
> Can someone explain why my list differs from the original? I am not
> understanding the hypothesis to generate the original list.
>
> Diana M.
>
> --
> "God made the integers, all else is the work of man."
> L. Kronecker, Jahresber. DMV 2, S. 19.




From g.resta at iit.cnr.it  Wed Jul 11 17:11:28 2007
From: g.resta at iit.cnr.it (Giovanni Resta)
Date: Wed, 11 Jul 2007 17:11:28 +0200
Subject: Question related to sequence A071267
In-Reply-To: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>
References: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>
Message-ID: <4694F320.3010100@iit.cnr.it>

Diana Mecum wrote:
> %I A071267
> %S A071267
> 2,4,6,8,10,11,12,14,16,18,22,33,44,55,66,77,88,99,101,110,121,132,143,
> %T A071267 154,165,176,187,198,111,222,666,888,1110
> %N A071267 Numbers which can be expressed as the sum of all distinct
> digit permutations of
>                some number k.
> When I try to follow the rule for generating the sequence numbers, I
> get the following list;
>
> 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110,
> 111, 121, 132, 143, 154, 165,
>  176, 187, 222, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443,
> 1554, 1665, 1776, 1887,
>  1998, 2109, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996
>
> Can someone explain why my list differs from the original? I am not
> understanding the hypothesis to generate the original list.
First of all, you must bear in mind that a consistent fraction of the
sequences submitted by Amarnath Murthy
are erroneous.

In any case it seems to me that also your list is not consistent with
the definition.
In particular, considering elements less or equal to 3996, you list
lacks the following
terms: 333, 1111, 2222, 2331, 2553, 2775, 3333
It is clear that all the repdigits (like 333, 1111, 2222, 3333, and so
on) belong to the
sequence since, for example, the set of all the distinct permutations of
333 is just 333.
(moreover 333 is also produced by the permutations of 300: 300+030+003 =
333).
The other missing values are produced as :
2331 = 399 + 993 + 939
2553 = 599+995+959
2775 = 799+997+979

So I think that the correct list, up to 10000, is
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
121, 132, 143, 154, 165, 176, 187, 222, 333, 444, 555, 666, 777, 888, 999,
1110, 1111, 1221, 1332, 1443, 1554, 1665, 1776, 1887, 1998, 2109, 2220,
2222,
2331, 2442, 2553, 2664, 2775, 2886, 3108, 3330, 3333, 3552, 3774, 3996,
4218,
4440, 4444, 4662, 4884, 5106, 5328, 5555, 6666, 7777, 8888, 9999

while, if we let drop the 'distinct' clause (so for example 11 produces
not 11 but 11+11=22), the
elements  up to 10000 are
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121,
132,
143, 154, 165, 176, 187, 198, 222, 444, 666, 888, 1110, 1332, 1554, 1776,
1998, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996, 4218, 4440,
4662,
4884, 5106, 5328, 5550, 5772, 5994, 6666

g.





From diana.mecum at gmail.com  Wed Jul 11 18:40:30 2007
From: diana.mecum at gmail.com (Diana Mecum)
Date: Wed, 11 Jul 2007 09:40:30 -0700
Subject: Question related to sequence A071267
In-Reply-To: <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com>
References: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>
	 <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com>
Message-ID: <a851af150707110940q2d6c38c1wf292f3ebe55e962@mail.gmail.com>

Joshua and Giovanni,

Thanks. I hadn't done a comprehensive initial list, having only looked at
200 terms. My question was more directed to why 1, 3, 5, 7 and 9 were not on
the list.

I will assume that my algorithm is correct, which agrees with your findings.

Thank you,

Diana M.


On 7/11/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
>
> Hi Diana and all,
> Murthy's sequences often have errors.  Here, among other problems, the
> definition is "numbers which ..." but then shouldn't they be SORTED?
>
> I don't quite agree with your list, either, though.  Here's what I
> think, based purely on brute-force with a computer, using as "seeds"
> all the numbers up to 100000:
>
> 1 2 3 4 5 6 7 8 9 11 22 33 44 55 66 77 88 99 110 111 121 132 143 154
> 165 176 187 222 333 444 555 666 777 888 999 1110 1111 1221 1332 1443
> 1554 1665 1776 1887 1998 2109 2220 2222 2331 2442 2553 2664 2775 2886
> 3108 3330 3333 3552 3774 3996 4218 4440 4444 4662 4884 5106 5328 5555
> 6666 7777 8888 9999 11110 11111 12221 13332 14443 15554 16665 17776
> 18887 19998 21109 22220 22222 23331 24442 25553 26664 27775 28886
> 29997
>
> which leaves me wondering, for instance, why I got 29997 but not 2997
> in there.  But now I see it's a nice little bit of combinatorics: 108
> goes to 1998 (six permutations, so effectively each spot is
> (1+0+8+1+0+8) so we get 1800 + 180 + 18), while 1008 goes to 29997
> (twelve permutations, so each spot is (1+0+0+8+1+0+0+8+1+0+0+8) so we
> get 27000 + 2700 + 270 + 27).
>
> So I think my above list of terms are correct.  Differences from your
> list, Diana, are that I have 333 and 1111 and 2222 and 2331 and 2553
> and 2775 and 3333 which are missing from your list.  (2331 for example
> comes from 399 -> 399 + 939 + 993, and any list of identical digits
> maps to itself, so 333 -> 333.)
>
> Diana, would you verify that you agree with my list of terms and then
> if you do please submit the corrected terms for this sequence?
>
> Thanks,
> --Joshua Zucker
>
>
> On 7/11/07, Diana Mecum <diana.mecum at gmail.com> wrote:
> > Sequence Fans,
> >
> > I am looking at sequence A071267.
> >
> > %I A071267
> > %S A071267
> > 2,4,6,8,10,11,12,14,16,18,22,33,44,55,66,77,88,99,101,110,121,132,143,
> > %T A071267 154,165,176,187,198,111,222,666,888,1110
> > %N A071267 Numbers which can be expressed as the sum of all distinct
> digit
> > permutations of
> >                some number k.
> > %C A071267 222 can be expressed so in two different ways i.e. 222= 200
> +020
> > + 002 as well as
> >                222= 101 +110 +011. Question: find a number which can be
> so
> > expressed
> >                 in n different ways.
> > %e A071267 1110 is a member as a sum of all distinct permutations of
> 104.
> > i.e. 104,140,410,
> >                401,014,041.
> > %Y A071267 Sequence in context: A081472 A097660 A067030 this_sequence
> > A072427 A050420 A096922
> > %Y A071267 Adjacent sequences: A071264 A071265 A071266 this_sequence
> A071268
> > A071269 A071270
> > %K A071267 base,more,nonn
> > %O A071267 1,1
> > %A A071267 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 01 2002
> >
> > When I try to follow the rule for generating the sequence numbers, I get
> the
> > following list;
> >
> > 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
> > 121, 132, 143, 154, 165,
> >  176, 187, 222, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443,
> 1554,
> > 1665, 1776, 1887,
> >  1998, 2109, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996
> >
> > Can someone explain why my list differs from the original? I am not
> > understanding the hypothesis to generate the original list.
> >
> > Diana M.
> >
> > --
> > "God made the integers, all else is the work of man."
> > L. Kronecker, Jahresber. DMV 2, S. 19.
>



-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From diana.mecum at gmail.com  Wed Jul 11 19:29:42 2007
From: diana.mecum at gmail.com (Diana Mecum)
Date: Wed, 11 Jul 2007 10:29:42 -0700
Subject: Question related to sequence A071267
In-Reply-To: <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com>
References: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>
	 <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com>
Message-ID: <a851af150707111029i23b67dcbgf2950a71f4be5d2c@mail.gmail.com>

Joshua,

Yes, I will double check that your and my terms agree. I will submit the
corrected list, as that was my purpose in studying the list in the first
place.

Thanks for your time.

Diana M.


On 7/11/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
>
> Hi Diana and all,
> Murthy's sequences often have errors.  Here, among other problems, the
> definition is "numbers which ..." but then shouldn't they be SORTED?
>
> I don't quite agree with your list, either, though.  Here's what I
> think, based purely on brute-force with a computer, using as "seeds"
> all the numbers up to 100000:
>
> 1 2 3 4 5 6 7 8 9 11 22 33 44 55 66 77 88 99 110 111 121 132 143 154
> 165 176 187 222 333 444 555 666 777 888 999 1110 1111 1221 1332 1443
> 1554 1665 1776 1887 1998 2109 2220 2222 2331 2442 2553 2664 2775 2886
> 3108 3330 3333 3552 3774 3996 4218 4440 4444 4662 4884 5106 5328 5555
> 6666 7777 8888 9999 11110 11111 12221 13332 14443 15554 16665 17776
> 18887 19998 21109 22220 22222 23331 24442 25553 26664 27775 28886
> 29997
>
> which leaves me wondering, for instance, why I got 29997 but not 2997
> in there.  But now I see it's a nice little bit of combinatorics: 108
> goes to 1998 (six permutations, so effectively each spot is
> (1+0+8+1+0+8) so we get 1800 + 180 + 18), while 1008 goes to 29997
> (twelve permutations, so each spot is (1+0+0+8+1+0+0+8+1+0+0+8) so we
> get 27000 + 2700 + 270 + 27).
>
> So I think my above list of terms are correct.  Differences from your
> list, Diana, are that I have 333 and 1111 and 2222 and 2331 and 2553
> and 2775 and 3333 which are missing from your list.  (2331 for example
> comes from 399 -> 399 + 939 + 993, and any list of identical digits
> maps to itself, so 333 -> 333.)
>
> Diana, would you verify that you agree with my list of terms and then
> if you do please submit the corrected terms for this sequence?
>
> Thanks,
> --Joshua Zucker
>
>
> On 7/11/07, Diana Mecum <diana.mecum at gmail.com> wrote:
> > Sequence Fans,
> >
> > I am looking at sequence A071267.
> >
> > %I A071267
> > %S A071267
> > 2,4,6,8,10,11,12,14,16,18,22,33,44,55,66,77,88,99,101,110,121,132,143,
> > %T A071267 154,165,176,187,198,111,222,666,888,1110
> > %N A071267 Numbers which can be expressed as the sum of all distinct
> digit
> > permutations of
> >                some number k.
> > %C A071267 222 can be expressed so in two different ways i.e. 222= 200
> +020
> > + 002 as well as
> >                222= 101 +110 +011. Question: find a number which can be
> so
> > expressed
> >                 in n different ways.
> > %e A071267 1110 is a member as a sum of all distinct permutations of
> 104.
> > i.e. 104,140,410,
> >                401,014,041.
> > %Y A071267 Sequence in context: A081472 A097660 A067030 this_sequence
> > A072427 A050420 A096922
> > %Y A071267 Adjacent sequences: A071264 A071265 A071266 this_sequence
> A071268
> > A071269 A071270
> > %K A071267 base,more,nonn
> > %O A071267 1,1
> > %A A071267 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 01 2002
> >
> > When I try to follow the rule for generating the sequence numbers, I get
> the
> > following list;
> >
> > 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
> > 121, 132, 143, 154, 165,
> >  176, 187, 222, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443,
> 1554,
> > 1665, 1776, 1887,
> >  1998, 2109, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996
> >
> > Can someone explain why my list differs from the original? I am not
> > understanding the hypothesis to generate the original list.
> >
> > Diana M.
> >
> > --
> > "God made the integers, all else is the work of man."
> > L. Kronecker, Jahresber. DMV 2, S. 19.
>



-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From davidwwilson at comcast.net  Thu Jul 12 07:41:32 2007
From: davidwwilson at comcast.net (David Wilson)
Date: Thu, 12 Jul 2007 01:41:32 -0400
Subject: Question related to sequence A071267
References: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com> <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com> <a851af150707111029i23b67dcbgf2950a71f4be5d2c@mail.gmail.com>
Message-ID: <000f01c7c447$4deb5920$6501a8c0@yourxhtr8hvc4p>


Let f(n) be the sum of all permutes of n.

Let

    s(n) = sum of digits of n.
    d(n) = number of digits of n.
    c_n(k) = number of occurences of digit k in n.
    p(n) = PROD(k = 0..9; c_n(k)!).
    r(n) = n-digit rep-1 number = (10^n-1)/n.
    t(n) = ((s(n)(d(n)-1)!)/p(n)).

Then f(n) = t(n) r(d(n)).


For example, if n = 314159, we get

    s(n) = 23
    d(n) = 6
    c_n = (0, 2, 0, 1, 1, 1, 0, 0, 0, 1)
    p(n) = PROD(k = 0..9; c_n(k)!) = 2
    r(d(n)) = r(6) = 111111
    t(n) = (23*120)/2 = 1380

and

    f(314159) = 1380*11111 = 153333180


A combinatorial argument shows that when we add the permutes of n, the same number of each digit will appear in each column of the sum. This means that

    f(n) = (sum of each column) * (d(n)-digit rep-1 number)
         = (sum of each column) * r(d(n))

from which we conclude

    t(n) = sum of each column

implying that t(n) is integer and r(d(n)) | f(n).


Using this knowledge, we can explore why f(n) = 29997 is insoluble while f(n) = 2997 is not.

Let n have d(n) digits. Then n <= 9 r(d(n)) and has at most d(n)! permutes. This means that

    [1] f(n) <= 9 r(d(n)) d(n)!

Suppose f(n) = 29997. We know that r(d(n)) | f(n), and the only rep-1 numbers that divide 29997 are r(1) = 1, r(2) = 11 and r(4) = 1111, implying d(n) = 1, 2 or 4. From [1],

    d(n) = 1  =>  f(n) <= 9
    d(n) = 2  =>  f(n) <= 198
    d(n) = 4  =>  f(n) <= 239976

Given f(n) = 29997, we must have d(n) = 4 and r(d(n)) = 1111.

This means that

    t(n) = f(n)/r(d(n)) = 29997/1111 = 27.

so that

    ((s(n)(d(n)-1)!)/p(n)) = 27.

Since d(n) = 4, we have

    6 s(n)/p(n) = 27
    => s(n) = 9(p(n)/2).

Since p(n) and s(n) are both integer, p(n) must be even to make the right side integer. This means 9 | s(n) <= 9 d(n) = 36. This means that s(n) = 9, 18, 27 or 36.

Suppose s(n) = 9. Then p(n) = 2. By the definition of p(n), p(n) = 2 iff exactly one digit appears twice in n and all others once (or not at all). So we are looking for a 4-digit n with a single repeated digit and digit sum 9. Many such numbers exist, 1008 being the smallest, and indeed f(1008) = 29997.


But now look at f(n) = 2997.  Following similar logic to the above, we find that d(n) = 3 and t(n) = 27 giving

    ((s(n)(d(n)-1)!)/p(n)) = 27
    => 2 s(n)/p(n) = 27
    => s(n) = 27(p(n)/2).

Again, p(n) must be even, so 27 | s(n). But s(n) <= 9 d(n) = 27, so s(n) = 27 forcing n = 999 and f(n) = 999. Hence f(n) = 2997 is insoluble.




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From njas at research.att.com  Thu Jul 12 10:45:29 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Thu, 12 Jul 2007 04:45:29 -0400 (EDT)
Subject: 3 seqs that should be the same but are not
In-Reply-To: <200707120845.l6C8jTGh29153108@fry.research.att.com>
References: <200707120845.l6C8jTGh29153108@fry.research.att.com>
Message-ID: <200707120845.l6C8jTGh29153108@fry.research.att.com>

Why should "surface area" (A074457) equal "volume" (A087300)?




From DWCantrell at sigmaxi.net  Thu Jul 12 15:01:07 2007
From: DWCantrell at sigmaxi.net (David W. Cantrell)
Date: Thu, 12 Jul 2007 14:01:07 +0100
Subject: 3 seqs that should be the same but are not
References: <200707120845.l6C8jTGh29153108@fry.research.att.com> <5542af940707120512v585336e2vb67179da5f641008@mail.gmail.com>
Message-ID: <001501c7c484$b79a7a30$25933e44@Dell>

----- Original Message ----- 
From: "Jonathan Post" <jvospost3 at gmail.com>
To: <njas at research.att.com>
Cc: <seqfan at ext.jussieu.fr>; <eric at weisstein.com>; <rgwv at rgwv.com>; 
"jonathan post" <jvospost2 at yahoo.com>
Sent: Thursday, July 12, 2007 13:12
Subject: Re: 3 seqs that should be the same but are not


> Why should "surface area" (A074457) equal "volume" (A087300)?

They're not the same constant, but the constant given in A074457 is 
simply 2 more than the constant given in A087300. Both entries should 
mention that, I think.

There are also two errors in the title of A087300. It should read:
Decimal expansion of value of d for which volume of d-dimensional unit 
ball is maximized.
[Note that it is necessary to specify that it's a _unit_ ball.]

David 





From DWCantrell at sigmaxi.net  Thu Jul 12 16:05:33 2007
From: DWCantrell at sigmaxi.net (David W. Cantrell)
Date: Thu, 12 Jul 2007 15:05:33 +0100
Subject: 3 seqs that should be the same but are not
References: <181140.23791.qm@web86606.mail.ukl.yahoo.com>
Message-ID: <001001c7c48d$b81894c0$25933e44@Dell>

Martin,

Concerning your question "Can someone explain why the dimension with 
maximum surface area is exactly 2 more than the dimension with maximum 
volume?":

On a "mechanical" level, explaining why is trivial. If we let invdig 
denote the appropriate inverse of the digamma function, then we can 
solve the required transcendental equations to obtain

2 invdig(log(pi))

and

2 invdig(log(pi)) + 2.

But you were probably wanting something deeper, and I'm not sure what 
that is.

David

----- Original Message ----- 
From: "Martin Fuller" <martin_n_fuller at btinternet.com>
To: "David W. Cantrell" <DWCantrell at sigmaxi.net>; "Jonathan Post" 
<jvospost3 at gmail.com>; <njas at research.att.com>
Cc: <seqfan at ext.jussieu.fr>; <eric at weisstein.com>; <rgwv at rgwv.com>; 
"jonathan post" <jvospost2 at yahoo.com>
Sent: Thursday, July 12, 2007 14:53
Subject: Re: 3 seqs that should be the same but are not


> A074455 and A087300 should be the same: A074455 has the right 
> description and
> came first, A087300 has the right values.
>
> I get a different value to OEIS for A074457 as well as for A074455.
>
> See below for the current values and my calculations in PARI.  The 
> PARI code
> gives the same value as A087300 and is consistent when increasing 
> the
> precision, so I think it's right.
>
> Martin Fuller
>
> PS Can someone explain why the dimension with maximum surface area 
> is exactly 2
> more than the dimension with maximum volume?
>
> PPS All these constants are based on geometers dimension which is 
> one more than
> topologists dimension (see the comments at MathWorld and compare 
> with the
> formulae at PlanetMath
> http://planetmath.org/encyclopedia/VolumeOfTheNSphere.html)
>
> OEIS
> A087300
> 5.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926
> A074455
> 5.25694640486057678013283838869076923661901723718321485750987966544135040807732427416016036408330066793834
> A074457
> 7.25694640486057678013283838869076923661901723718321485750987966581295771725141939748631782716397233020121
> A074454
> 5.27776802111340099728214586417284638752999928451017356776163734021486412730547017110062048407258401284645
> A074456
> 33.1611944849620026918630240155829735800472328410872585131001181554037565464718434466607460949351387694776
>
> PARI/GP
> A074455
> 5.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926957
> A074457
> 7.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926957
> A074454
> 5.27776802111340099728214586417284638752999928451017356776163734021486412730547017110062048407258401284645
> A074456
> 33.1611944849620026918630240155829735800472328410872585131001181554037565464718434466607460949351387694776
>
> /* PARI/GP code */
> hyperspheresurface(d)=2*Pi^(d/2)/gamma(d/2)
> hyperspherevolume(d)=hyperspheresurface(d)/d
>
> FindMax(fn_x,lo,hi)=
> {
>  local(oldprecision, x, y, z);
>  oldprecision = default(realprecision);
>  default(realprecision, oldprecision+10);
>
>  while (hi-lo > 10^-oldprecision,
>    while (1,
>      z = vector(2, i, lo*(3-i)/3 + hi*i/3);
>      y = vector(2, i, eval(Str("x = z[" i "]; " fn_x)));
>      if (abs(y[1]-y[2]) > 10^(5-default(realprecision)), break);
>      default(realprecision, default(realprecision)+10);
>    );
>    if (y[1] < y[2], lo = z[1], hi = z[2]);
>  );
>
>  default(realprecision, oldprecision);
>  (lo + hi) / 2.
> }
>
> default(realprecision, 105);
> A074455=FindMax("hyperspherevolume(x)", 1, 9)
> A074457=FindMax("hyperspheresurface(x)", 1, 9)
> A074454=hyperspherevolume(A074455)
> A074456=hyperspheresurface(A074457)
> /* PARI/GP code ends */
>
> --- "David W. Cantrell" <DWCantrell at sigmaxi.net> wrote:
>
>> ----- Original Message ----- 
>> From: "Jonathan Post" <jvospost3 at gmail.com>
>> To: <njas at research.att.com>
>> Cc: <seqfan at ext.jussieu.fr>; <eric at weisstein.com>; <rgwv at rgwv.com>;
>> "jonathan post" <jvospost2 at yahoo.com>
>> Sent: Thursday, July 12, 2007 13:12
>> Subject: Re: 3 seqs that should be the same but are not
>>
>>
>> > Why should "surface area" (A074457) equal "volume" (A087300)?
>>
>> They're not the same constant, but the constant given in A074457 is
>> simply 2 more than the constant given in A087300. Both entries 
>> should
>> mention that, I think.
>>
>> There are also two errors in the title of A087300. It should read:
>> Decimal expansion of value of d for which volume of d-dimensional 
>> unit
>> ball is maximized.
>> [Note that it is necessary to specify that it's a _unit_ ball.]
>>
>> David
>>
>> 





From diana.mecum at gmail.com  Thu Jul 12 16:36:00 2007
From: diana.mecum at gmail.com (Diana Mecum)
Date: Thu, 12 Jul 2007 09:36:00 -0500
Subject: Question related to sequence A071267
In-Reply-To: <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com>
References: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>
	 <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com>
Message-ID: <a851af150707120736o7e035362t73e6c2a4bc08914e@mail.gmail.com>

Joshua,

Performing permutations on numbers up to 100,000 gave me the following
numbers to add to sequence A071267. I will add these to the sequence. Diana
M.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
121, 132, 143, 154, 165,
 176, 187, 222, 333, 444, 555, 666, 777, 888, 999, 1110, 1111, 1221, 1332,
1443, 1554, 1665, 1776,
 1887, 1998, 2109, 2220, 2222, 2331, 2442, 2553, 2664, 2775, 2886, 3108,
3330, 3333, 3552, 3774,
 3996, 4218, 4440, 4444, 4662, 4884, 5106, 5328, 5555, 6666, 7777, 8888,
9999, 11110, 11111,
 12221, 13332, 14443, 15554, 16665, 17776, 18887, 19998, 21109, 22220,
22222, 23331, 24442, 25553,
 26664, 27775, 28886, 29997, 31108, 32219, 33330, 33333, 34441, 35552,
36663, 37774, 38885, 39996,
 43329, 44444, 46662, 49995, 53328, 55555, 56661, 59994, 63327, 66660,
66666, 69993, 73326, 76659,
 77777, 79992, 83325, 86658, 88888, 89991, 93324, 96657, 99990, 99999,
103323, 106656, 109989,
 111110, 111111, 113322, 119988, 122221, 126654, 133320, 133332, 139986,
144443, 146652, 153318,
 155554, 159984, 166650, 166665, 173316, 177776, 179982, 186648, 188887,
193314, 199980, 199998,
 211109, 222220, 233331, 244442, 255553, 266664, 277775, 288886, 299997,
311108, 322219, 333330,
 344441, 355552, 366663, 377774, 388885, 399996, 411107, 422218, 433329,
444440, 455551, 466662,
 477773, 488884, 511106, 533328, 555550, 577772, 599994, 622216, 644438,
666660, 688882, 711104,
 733326, 755548, 777770, 799992, 822214, 844436, 866658, 888880, 911102,
933324, 955546, 977768,
 999990, 1022212, 1066656, 1111100, 1133322, 1155544, 1199988, 1244432,
1266654, 1288876, 1333320,
 1377764, 1399986, 1422208, 1466652, 1511096, 1533318, 1555540, 1599984,
1644428, 1666650,
 1688872, 1733316, 1777760, 1799982, 1822204, 1866648, 1933314, 1999980,
2066646, 2133312,
 2199978, 2266644, 2333310, 2399976, 2466642, 2533308, 2599974, 2666640,
2733306, 2799972,
 2933304, 3066636, 3199968, 3333300, 3466632, 3599964, 3733296, 3866628,
3999960, 4133292,
 4266624, 4399956, 4533288, 4666620, 4799952, 4933284, 5066616, 5199948,
5333280, 5599944,
 5866608, 6133272, 6399936, 6666600, 6933264, 7199928, 7466592, 7733256,
7999920, 8266584,
 8533248, 8799912, 9066576, 9333240

On 7/11/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
>
> Hi Diana and all,
> Murthy's sequences often have errors.  Here, among other problems, the
> definition is "numbers which ..." but then shouldn't they be SORTED?
>
> I don't quite agree with your list, either, though.  Here's what I
> think, based purely on brute-force with a computer, using as "seeds"
> all the numbers up to 100000:
>
> 1 2 3 4 5 6 7 8 9 11 22 33 44 55 66 77 88 99 110 111 121 132 143 154
> 165 176 187 222 333 444 555 666 777 888 999 1110 1111 1221 1332 1443
> 1554 1665 1776 1887 1998 2109 2220 2222 2331 2442 2553 2664 2775 2886
> 3108 3330 3333 3552 3774 3996 4218 4440 4444 4662 4884 5106 5328 5555
> 6666 7777 8888 9999 11110 11111 12221 13332 14443 15554 16665 17776
> 18887 19998 21109 22220 22222 23331 24442 25553 26664 27775 28886
> 29997
>
> which leaves me wondering, for instance, why I got 29997 but not 2997
> in there.  But now I see it's a nice little bit of combinatorics: 108
> goes to 1998 (six permutations, so effectively each spot is
> (1+0+8+1+0+8) so we get 1800 + 180 + 18), while 1008 goes to 29997
> (twelve permutations, so each spot is (1+0+0+8+1+0+0+8+1+0+0+8) so we
> get 27000 + 2700 + 270 + 27).
>
> So I think my above list of terms are correct.  Differences from your
> list, Diana, are that I have 333 and 1111 and 2222 and 2331 and 2553
> and 2775 and 3333 which are missing from your list.  (2331 for example
> comes from 399 -> 399 + 939 + 993, and any list of identical digits
> maps to itself, so 333 -> 333.)
>
> Diana, would you verify that you agree with my list of terms and then
> if you do please submit the corrected terms for this sequence?
>
> Thanks,
> --Joshua Zucker
>
>
> On 7/11/07, Diana Mecum <diana.mecum at gmail.com> wrote:
> > Sequence Fans,
> >
> > I am looking at sequence A071267.
> >
> > %I A071267
> > %S A071267
> > 2,4,6,8,10,11,12,14,16,18,22,33,44,55,66,77,88,99,101,110,121,132,143,
> > %T A071267 154,165,176,187,198,111,222,666,888,1110
> > %N A071267 Numbers which can be expressed as the sum of all distinct
> digit
> > permutations of
> >                some number k.
> > %C A071267 222 can be expressed so in two different ways i.e. 222= 200
> +020
> > + 002 as well as
> >                222= 101 +110 +011. Question: find a number which can be
> so
> > expressed
> >                 in n different ways.
> > %e A071267 1110 is a member as a sum of all distinct permutations of
> 104.
> > i.e. 104,140,410,
> >                401,014,041.
> > %Y A071267 Sequence in context: A081472 A097660 A067030 this_sequence
> > A072427 A050420 A096922
> > %Y A071267 Adjacent sequences: A071264 A071265 A071266 this_sequence
> A071268
> > A071269 A071270
> > %K A071267 base,more,nonn
> > %O A071267 1,1
> > %A A071267 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 01 2002
> >
> > When I try to follow the rule for generating the sequence numbers, I get
> the
> > following list;
> >
> > 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
> > 121, 132, 143, 154, 165,
> >  176, 187, 222, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443,
> 1554,
> > 1665, 1776, 1887,
> >  1998, 2109, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996
> >
> > Can someone explain why my list differs from the original? I am not
> > understanding the hypothesis to generate the original list.
> >
> > Diana M.
> >
> > --
> > "God made the integers, all else is the work of man."
> > L. Kronecker, Jahresber. DMV 2, S. 19.
>



-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From martin_n_fuller at btinternet.com  Thu Jul 12 15:53:02 2007
From: martin_n_fuller at btinternet.com (Martin Fuller)
Date: Thu, 12 Jul 2007 14:53:02 +0100 (BST)
Subject: 3 seqs that should be the same but are not
In-Reply-To: <001501c7c484$b79a7a30$25933e44@Dell>
Message-ID: <181140.23791.qm@web86606.mail.ukl.yahoo.com>

A074455 and A087300 should be the same: A074455 has the right description and
came first, A087300 has the right values.

I get a different value to OEIS for A074457 as well as for A074455.

See below for the current values and my calculations in PARI.  The PARI code
gives the same value as A087300 and is consistent when increasing the
precision, so I think it's right.

Martin Fuller

PS Can someone explain why the dimension with maximum surface area is exactly 2
more than the dimension with maximum volume?

PPS All these constants are based on geometers dimension which is one more than
topologists dimension (see the comments at MathWorld and compare with the
formulae at PlanetMath
http://planetmath.org/encyclopedia/VolumeOfTheNSphere.html)

OEIS
A087300
5.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926
A074455
5.25694640486057678013283838869076923661901723718321485750987966544135040807732427416016036408330066793834
A074457
7.25694640486057678013283838869076923661901723718321485750987966581295771725141939748631782716397233020121
A074454
5.27776802111340099728214586417284638752999928451017356776163734021486412730547017110062048407258401284645
A074456
33.1611944849620026918630240155829735800472328410872585131001181554037565464718434466607460949351387694776

PARI/GP 
A074455
5.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926957
A074457
7.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926957
A074454
5.27776802111340099728214586417284638752999928451017356776163734021486412730547017110062048407258401284645
A074456
33.1611944849620026918630240155829735800472328410872585131001181554037565464718434466607460949351387694776

/* PARI/GP code */
hyperspheresurface(d)=2*Pi^(d/2)/gamma(d/2)
hyperspherevolume(d)=hyperspheresurface(d)/d

FindMax(fn_x,lo,hi)=
{
  local(oldprecision, x, y, z);
  oldprecision = default(realprecision);
  default(realprecision, oldprecision+10);

  while (hi-lo > 10^-oldprecision,
    while (1,
      z = vector(2, i, lo*(3-i)/3 + hi*i/3);
      y = vector(2, i, eval(Str("x = z[" i "]; " fn_x)));
      if (abs(y[1]-y[2]) > 10^(5-default(realprecision)), break);
      default(realprecision, default(realprecision)+10);
    );
    if (y[1] < y[2], lo = z[1], hi = z[2]);
  );

  default(realprecision, oldprecision);
  (lo + hi) / 2.
}

default(realprecision, 105);
A074455=FindMax("hyperspherevolume(x)", 1, 9)
A074457=FindMax("hyperspheresurface(x)", 1, 9)
A074454=hyperspherevolume(A074455)
A074456=hyperspheresurface(A074457)
/* PARI/GP code ends */

--- "David W. Cantrell" <DWCantrell at sigmaxi.net> wrote:

> ----- Original Message ----- 
> From: "Jonathan Post" <jvospost3 at gmail.com>
> To: <njas at research.att.com>
> Cc: <seqfan at ext.jussieu.fr>; <eric at weisstein.com>; <rgwv at rgwv.com>; 
> "jonathan post" <jvospost2 at yahoo.com>
> Sent: Thursday, July 12, 2007 13:12
> Subject: Re: 3 seqs that should be the same but are not
> 
> 
> > Why should "surface area" (A074457) equal "volume" (A087300)?
> 
> They're not the same constant, but the constant given in A074457 is 
> simply 2 more than the constant given in A087300. Both entries should 
> mention that, I think.
> 
> There are also two errors in the title of A087300. It should read:
> Decimal expansion of value of d for which volume of d-dimensional unit 
> ball is maximized.
> [Note that it is necessary to specify that it's a _unit_ ball.]
> 
> David 
> 
> 





From maximilian.hasler at gmail.com  Fri Jul 13 04:16:00 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Thu, 12 Jul 2007 22:16:00 -0400
Subject: link to 3n+1 ?
Message-ID: <3c3af2330707121916p1f22271bxb9b1a6eac4a04415@mail.gmail.com>

%I A001281
%S A001281 2,1,8,2,14,3,20,4,26,5,32,6,38,7,44,8,50,9,56,10,62,11,68,12,74,13,80,
%N A001281 Image of n under n->n/2 if n even, n->3n-1 if n odd.
%Y A001281 Cf. A037082.

I may be wrong but at first glance, the xref given in this record
refers to something completely unrelated(?) (Primes of the form n!!! +
1),
shouldn't it rather refer to

A006370  Image of n under the `3x+1' map.

or the like? (I could not spot an obvious typo comparing with id's at
http://www.research.att.com/~njas/sequences/Sindx_3.html#3x1

On that token,

%S A008908 1,2,8,3,6,9,17,4,20,7,15,10,10,18,18
%N A008908 Number of halving and tripling steps to reach 1 in `3x+1' problem.
%Y A008908 a(n) = A006577(n) + 1.

seems inconsistent with

%S A006577 0,1,7,2,5,8,16,3,19,6,14,9,9,17,17,4,12,
%N A006577 Number of halving and tripling steps to reach 1 in `3x+1' problem.

(the %N should not be identical if values differ)

M.H.



I've now added all the sequences that were submitted
to this point in time, including those that were "lost".

[The "lost" ones have numbers in the range A130707 - A130742.
There may be some duplicates there - let me know if you notice
any.  I checked, but not very thoroughly.  Getting the "lost" ones
out of the log files was a lot of work.]

The next project is to process "Comments".  There are about 500,

In the meantime, please remember that the OEIS is on vacation.

Neil




From njas at research.att.com  Fri Jul 13 16:12:16 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Fri, 13 Jul 2007 10:12:16 -0400 (EDT)
Subject: OEIS status report
Message-ID: <200707131412.l6DECGEp29694119@fry.research.att.com>

so this will take a while.
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Message-ID: <335cf8920707131046v25a23917s3adf511e259b60c at mail.gmail.com>
Date: Fri, 13 Jul 2007 19:46:07 +0200
From: "Olivier Gerard" <ogerard at ext.jussieu.fr>
Sender: olivier.gerard at gmail.com
To: seqfan at ext.jussieu.fr
Subject: [seqfan] Online again
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Dear Members,
As the last few messages have proven, the mailing listserver for SeqFan is up again and in good condition.
Have a pleasant day.
Olivier GERARD
====================================
Chers abonn??s de la liste seqfan,
Comme les quelques messages r??cents l'ont montr??,le syst??me informatique de notre liste de discussionest de nouveau op??rationnel.
A tous une bonne journ??e,
Olivier GERARD




From nholmes at leven.comp.utas.edu.au  Sat Jul 14 13:38:18 2007
From: nholmes at leven.comp.utas.edu.au (Neville Holmes)
Date: Sat, 14 Jul 2007 21:38:18 +1000 (EST)
Subject: A008776 and A025192 Duplicates ?
Message-ID: <20070714113818.9533E106B5@leven.comp.utas.edu.au>

A008776 and A025192 look the same to me.  Am I missing something ?
-------------------------------------------------------------------------
Neville Holmes                 E-mail: Neville.Holmes at utas.edu.au
School of Computing            Phone : +61 363 243 393 or (03) 63 243 393
University of Tasmania         Fax   : +61 363 243 368 or (03) 63 243 368
Locked Bag 1-359  Launceston 7250 ---------------------------------------



Neville,  They are not quite the same - one has an 
initial 1.  The relation between them is explicitly mentioned
in the entries.

Neil




From njas at research.att.com  Sat Jul 14 15:28:12 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Sat, 14 Jul 2007 09:28:12 -0400 (EDT)
Subject: A008776 and A025192 Duplicates ?
Message-ID: <200707141328.l6EDSCSj27853097@fry.research.att.com>



A124502 is essentially the same as A108458?


http://www.research.att.com/~njas/sequences/A108458

http://www.research.att.com/~njas/sequences/A124502

V.
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From aomunagi at gmail.com  Mon Jul 16 17:52:58 2007
From: aomunagi at gmail.com (Augustine Munagi)
Date: Mon, 16 Jul 2007 17:52:58 +0200
Subject: Possible duplicate
In-Reply-To: <102c01c7c6d4$a4b992a0$3701f0d5@speedy>
References: <102c01c7c6d4$a4b992a0$3701f0d5@speedy>
Message-ID: <e1c0995c0707160852t74d789b8yce96e6910d7cbc6a@mail.gmail.com>

Hi Seqfans,

They should not be the same unless the definition of A108458 is
modified. A108458 needs improvement.
They are not transposes, even though A108458(n,k)=A124502(n+k,k+1) for
fixed k. By definition A108458 is infinite both downwards and to the
right while A124502 is finite to the right.



On 7/15/07, Vladeta Jovovic <vladeta at eunet.yu> wrote:
>
>
>
> A124502 is essentially the same as A108458?
>
> http://www.research.att.com/~njas/sequences/A108458
>
> http://www.research.att.com/~njas/sequences/A124502
>
> V.




From maxale at gmail.com  Tue Jul 17 06:20:25 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Mon, 16 Jul 2007 21:20:25 -0700
Subject: A091902 is duplicate of A067698
Message-ID: <d3dac270707162120j4d5d510bhe25aa99d85a2a8b2@mail.gmail.com>

SeqFan

A091902 is a duplicate of A067698. They differ only with an extra
first term 1 in A091902, which actually should not be there since
log(log(n)) in not defined for n=1.
A067698 is much developed than A091902, missing only a link to Robin's
Theorem. So, I suggest to add a link to Robin's Theorem to A067698 and
remove A091902 from OEIS.

Max



I have recently submitted these sequences, which have already appeared in 
the database.

>%I A131789
>%S A131789 1,2,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,2,1,1,1,1,1,3,1
>%N A131789 a(n) = the length of the nth run of similar consecutive values 
>in the sequence A000005. (A000005(n) = the number of positive divisors of n.)
>%e A131789 Runs of sequence A000005: (1), (2,2), (3), (2), (4), (2), (4), 
>(3), (4), (2), (6), (2), (4,4), (5), (2), (6), (2), (6), (4,4), (2), (8), 
>(3),...
>%Y A131789 A000005,A131790
>%O A131789 1
>%K A131789 ,more,nonn,


>%I A131790
>%S A131790 1,1,10,1,5,1,3,1,5,1
>%N A131790 a(n) = the length of the nth run of similar consecutive values 
>in the sequence A131789.
>%e A131790 Runs of sequence A131789: (1), (2), (1,1,1,1,1,1,1,1,1,1), (2), 
>(1,1,1,1,1), (2), (1,1,1), (2), (1,1,1,1,1,), (3),...
>%Y A131790 A000005,A131789
>%O A131790 1
>%K A131790 ,more,nonn,


It seems VERY likely to me that there is no infinite string of 1's, or of 
anything else, in sequence A131789 (ie. the terms of A131790 are all 
finite).

Can it be PROVED that all terms of A131790 are finite, possibly using 
Hardy and Wright or some other such reference?

A harder question to answer:
We can define sequence S(m) = {s(m,n)}, where s(m,n) = the length of the 
nth run of similar consecutive values in the sequence S(m-1), where S(0) 
= sequence A000005.
(And S(1) = A131789, S(2) = A131790, of course.)

Is every term of S(m) finite for every m = positive integer?

It seems intuitive obvious that, yes, all terms are finite. But a proof 
would be harder to produce.


Thanks,
Leroy Quet




From qq-quet at mindspring.com  Tue Jul 17 16:45:59 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Tue, 17 Jul 07 08:45:59 -0600
Subject: Lengths Of Runs In The #-Of-Divisors Sequence
Message-ID: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>


Just for fun (?):
http://www.cetteadressecomportecinquantesignes.com/WordsPosition.htm
Best,
?.





From joshua.zucker at gmail.com  Tue Jul 17 21:44:10 2007
From: joshua.zucker at gmail.com (Joshua Zucker)
Date: Tue, 17 Jul 2007 12:44:10 -0700
Subject: Lengths Of Runs In The #-Of-Divisors Sequence
In-Reply-To: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
References: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
Message-ID: <721e81490707171244r38763dedj89e40b5e5d7bee5b@mail.gmail.com>

On 7/17/07, Leroy Quet <qq-quet at mindspring.com> wrote:
> I have recently submitted these sequences, which have already appeared in
> the database.

Should be easy to extend -- is it easier to post here first, asking
someone with a computer to extend them for you, and then submit enough
terms to begin with?

> It seems VERY likely to me that there is no infinite string of 1's, or of
> anything else, in sequence A131789 (ie. the terms of A131790 are all
> finite).
>
> Can it be PROVED that all terms of A131790 are finite, possibly using
> Hardy and Wright or some other such reference?

No need for Hardy and Wright, just Euclid -- there are infinitely many
primes.  Plus there are not any consecutive primes after 2.  Hence the
primes (2s) partition the list into finite chunks.

--Joshua Zucker




From joshua.zucker at gmail.com  Tue Jul 17 21:45:14 2007
From: joshua.zucker at gmail.com (Joshua Zucker)
Date: Tue, 17 Jul 2007 12:45:14 -0700
Subject: Lengths Of Runs In The #-Of-Divisors Sequence
In-Reply-To: <721e81490707171244r38763dedj89e40b5e5d7bee5b@mail.gmail.com>
References: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
	 <721e81490707171244r38763dedj89e40b5e5d7bee5b@mail.gmail.com>
Message-ID: <721e81490707171245r7865a036ude11b441e53b8ae7@mail.gmail.com>

On 7/17/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
> On 7/17/07, Leroy Quet <qq-quet at mindspring.com> wrote:

> > It seems VERY likely to me that there is no infinite string of 1's, or of
> > anything else, in sequence A131789 (ie. the terms of A131790 are all
> > finite).
> >
> > Can it be PROVED that all terms of A131790 are finite, possibly using
> > Hardy and Wright or some other such reference?
>
> No need for Hardy and Wright, just Euclid -- there are infinitely many
> primes.  Plus there are not any consecutive primes after 2.  Hence the
> primes (2s) partition the list into finite chunks.

Wups, I think that's a proof for A131789, not for A131790.

--Joshua



Hello all,

I was looking at A069803 - Smaller of two consecutive palindromic primes:	2, 3, 5, 7, 181, 787, 919 
Conjectured to be complete. 

I am interested in seeing a proof that 919 is actually the largest palindromic prime such that the next prime is palindromic. 
I checked up to 10^8 with Mathematica coding. 

Also, it is obvious that the distance from a palindrome n to the next one is more than Sqrt(n/10). It is clear that prime gaps grow slower than that. Looking at the prime gaps sequence A053303, it is easy to prove that 919 is the last number like that up to 10^16.

Is there a bound for prime gaps that proves that the gaps are less than Sqrt(n/10) starting from some n?

Tanya


_________________________________________________________________
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From tanyakh at TanyaKhovanova.com  Wed Jul 18 00:12:20 2007
From: tanyakh at TanyaKhovanova.com (Tanya Khovanova)
Date: Tue, 17 Jul 2007 15:12:20 -0700
Subject: 919 conjecture
In-Reply-To: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
References: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
Message-ID: <200707171512.AA1076494578@TanyaKhovanova.com>

On 7/17/07, Leroy Quet <qq-quet at mindspring.com> wrote:

> Can it be PROVED that all terms of A131790 are finite, possibly using
> Hardy and Wright or some other such reference?

It would follow from the statement:

There exist infinitely many positive integers k such that both 2k+1
and 3k+1 are prime.
I do not have a proof that this statement is true (it may be as hard
as twin prime conjecture) but heuristic arguments ala Hardy--Wright
suggest that it is ture.

Now, if for some k>1 both 2k+1 and 3k+1 are prime then 3(2k+1)=6k+3
and 2(3k+1)=6k+2 both have exactly 4 divisors, implying that
A000005(6k+3)=A000005(6k+2)=4 belong to the same run in A000005 and
the length of this run is greater than 1. In other words, in A131789
elements greater than 1 appear infinitely often.
It is also clear that for each prime p >= 5, A000005(p)=2 forms its
own run of length 1 in A000005, implying that in A131789 elements
equal 1 appear infinitely often.
Therefore, all runs in A131789 are finite and so are elements of A131790.

P.S. btw, the sequence of k such that both 2k+1 and 3k+1 are prime
seems to be missing in OEIS. It starts with:
2, 6, 14, 20, 26, 36, 50, 54, 74, 90, 116, 140, 146, 174, 200, 204,
210, 224, 230, 270, 284, 306, 330, 336, 350, 354, 384, 404, 410, 426,
440, 476, 510, 516, 554, 564, 596, 600, 624, 644, 650, 704, 714, 726,
740, 746, 834, 846, 894, 930, 944, 950

Regards,
Max




From franktaw at netscape.net  Wed Jul 18 05:21:12 2007
From: franktaw at netscape.net (franktaw at netscape.net)
Date: Tue, 17 Jul 2007 23:21:12 -0400
Subject: 919 conjecture
In-Reply-To: <200707171512.AA1076494578@TanyaKhovanova.com>
References: <200707171512.AA1076494578@TanyaKhovanova.com>
Message-ID: <8C99701D3845E00-E70-2A8D@FWM-D43.sysops.aol.com>

No.

It is an open problem of long standing to show that there is always a 
prime between n^2 and (n+1)^2.  This means that we can't prove that 
prime gaps are always less than sqrt(n) for n sufficiently large.

Franklin T. Adams-Watters

-----Original Message-----
From: Tanya Khovanova <tanyakh at TanyaKhovanova.com>

Hello all,

I was looking at A069803 - Smaller of two consecutive palindromic 
primes:   2,
3, 5, 7, 181, 787, 919
Conjectured to be complete.

I am interested in seeing a proof that 919 is actually the largest 
palindromic
prime such that the next prime is palindromic.
I checked up to 10^8 with Mathematica coding.

Also, it is obvious that the distance from a palindrome n to the next 
one is
more than Sqrt(n/10). It is clear that prime gaps grow slower than 
that. Looking
at the prime gaps sequence A053303, it is easy to prove that 919 is the 
last
number like that up to 10^16.

Is there a bound for prime gaps that proves that the gaps are less than
Sqrt(n/10) starting from some n?

Tanya



________________________________________________________________________
Check Out the new free AIM(R) Mail -- Unlimited storage and 
industry-leading spam and email virus protection.




From aomunagi at gmail.com  Wed Jul 18 17:16:54 2007
From: aomunagi at gmail.com (Augustine Munagi)
Date: Wed, 18 Jul 2007 17:16:54 +0200
Subject: Possible duplicate
In-Reply-To: <e1c0995c0707160852t74d789b8yce96e6910d7cbc6a@mail.gmail.com>
References: <102c01c7c6d4$a4b992a0$3701f0d5@speedy>
	 <e1c0995c0707160852t74d789b8yce96e6910d7cbc6a@mail.gmail.com>
Message-ID: <e1c0995c0707180816v27f11abj37566c3304af1ec6@mail.gmail.com>

Hi Seqfan,
Additionally, the relation A108458(n,k)=A124502(n+k,k+1) shows that
the sequences are related like "multichoose" to "choose". I suggest a
merging of the sequences by inserting the definition of A108458 as a
comment in A124502 with the above relation.
Else A108458 as presently displayed is inaccurate, and indeed leaves
us with a duplicate.



On 7/16/07, Augustine Munagi <aomunagi at gmail.com> wrote:
> Hi Seqfans,
>
> They should not be the same unless the definition of A108458 is
> modified. A108458 needs improvement.
> They are not transposes, even though A108458(n,k)=A124502(n+k,k+1) for
> fixed k. By definition A108458 is infinite both downwards and to the
> right while A124502 is finite to the right.
>
>
>
> On 7/15/07, Vladeta Jovovic <vladeta at eunet.yu> wrote:
> >
> >
> >
> > A124502 is essentially the same as A108458?
> >
> > http://www.research.att.com/~njas/sequences/A108458
> >
> > http://www.research.att.com/~njas/sequences/A124502
> >
> > V.
>




From joshua.zucker at gmail.com  Wed Jul 18 23:19:32 2007
From: joshua.zucker at gmail.com (Joshua Zucker)
Date: Wed, 18 Jul 2007 14:19:32 -0700
Subject: Lengths Of Runs In The #-Of-Divisors Sequence
In-Reply-To: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
References: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
Message-ID: <721e81490707181419y19dc932fy51af3d71b8b0cf0@mail.gmail.com>

On 7/17/07, Leroy Quet <qq-quet at mindspring.com> wrote:
> A harder question to answer:
> We can define sequence S(m) = {s(m,n)}, where s(m,n) = the length of the
> nth run of similar consecutive values in the sequence S(m-1), where S(0)
> = sequence A000005.
> (And S(1) = A131789, S(2) = A131790, of course.)
>
> Is every term of S(m) finite for every m = positive integer?
>
> It seems intuitive obvious that, yes, all terms are finite. But a proof
> would be harder to produce.

There seems to be a strongly alternating pattern:
Divisor sequence: 1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6 4 4 2 8 3 4
4 6 2 8 2 6 4 4 4 9 2 4 4 8 2 8 2 6 6 4 2 10 3 6 4 6 2 8 4 8 4 4 2 12
2 4 6 7 4 8 2 6 4 8 2 12 2 4 6 6 4 8 2 10 5 4 2 12 4 4 4 8 2 12 4 6 4
4 4 12 2 6 6 9

Run lengths in divisor sequence: 1 2 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2
1 1 1 2 1 1 1 1 1 3 1 1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 3 1 1 1 1 1 3 1 1 2 1 1 1 1 2
1 1 1 1 1 1 1 1 1 1 2

Run lengths in that: 1 1 10 1 5 1 3 1 5 1 2 1 4 1 11 1 16 1 8 1 5 1 2
1 4 1 10 2 2 1 9 2 4 1 1 2 9 1 11 1 4 1 10 1 10 1 1 1 6 1 1 1 10 1 9 1
7 1 30 1 9 2 1 1 22 1 4 2 8 1 28 1 4 1 4 1 4 1 33 1 3 1 9 1 5 1 26 1
18 1 4 1 5 1 10 1 9 1 3 1

Run lengths in that: 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 3 1 3 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1

And then: 1 25 1 4 1 10 1 1 1 10 1 55 1 15 1 4 1 63 1 3 1 12 1 33 1 5
1 32 1 9 1 7 1 3 1 13 1 56 1 61 1 5 1 103 1 47 1 17 1 13 1 25 1 5 1 5
1 47 1 3 1 21 1 7 1 11 1 1 1 17 1 3 1 1 1 8 1 1 1 5 1 7 1 9 1 2 1 15 1
36 1 5 1 11 1 7 1 1 1 15

And then: 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3
1 1 1 3 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1

And then: 6 1 57 1 3 1 1 1 17 1 63 1 7 1 9 1 101 1 17 1 43 1 13 1 27 1
129 1 37 1 17 1 9 1 39 1 15 1 15 1 45 1 27 1 11 1 3 1 19 1 41 1 5 1 51
1 9 1 47 1 33 1 15 1 35 1 7 1 7 1 15 1 13 1 29 1 53 1 25 1 9 1 23 1 69
1 9 1 5 1 35 1 13 1 9 1 9 1 37 1

And then: 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

And then: 5 1 105 1 3 1 49 1 29 1 5 1 31 1 15 1 9 1 15 1 17 1 5 1 61 1
9 1 1 1 1 1 1 1 13 1 11 1 11 1 41 1 51 1 19 1 73 1 5 1 55 1 31 1 41 1
67 1 133 1 5 1 5 1 85 1 9 1 7 1 61 1 11 1 5 1 1 1 15 1 29 1

And so on, where alternately you get (long strings of 1s with the
occasional other number) and (alternately 1s and other numbers, with
occasionally 2 or 3 1s in a row).

By the way, using the first million terms of A000005, I get only the
first 82 terms of S(8), so it'll take some work to explore this any
further numerically.

I don't think it's at all obvious that this thing won't eventually
yield a sequence of all 1s, even if it is intuitively obvious to Leroy
Quet, it's sure not obvious to me!

--Joshua Zucker
PS: I also submitted extensions of A131789 and A131790 as part of this work.



Joshua Zucker wrote:
>I don't think it's at all obvious that this thing won't eventually
>yield a sequence of all 1s, even if it is intuitively obvious to Leroy
>Quet, it's sure not obvious to me!

I think my use of the phrase "intuitively obvious" is quite an 
exaggeration.
It just seems *likely* to me that the terms of each S(m) are finite. 
Nothing is really obvious.

:)

Thanks,
Leroy Quet




From qq-quet at mindspring.com  Wed Jul 18 23:44:20 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Wed, 18 Jul 07 15:44:20 -0600
Subject: Lengths Of Runs In The #-Of-Divisors Sequence
Message-ID: <E1IBHMR-00036C-00@pop05.mail.atl.earthlink.net>

This is a multi-part message in MIME format.

------_=_NextPart_001_01C7CA44.3128B80C
Content-Type: text/plain;
	charset="iso-8859-1"
Content-Transfer-Encoding: quoted-printable

Hello SeqFans,
could someone check and compute a hundred terms or so of the seq below?
=20
4,6,9,21,22,25,33,39,46,49,51,54,...
=20
Integers n whose "ordered concatenation" of all divisors (except 1 and n) i=
s a prime.
=20
 n     div.         prime
 4 -> 1,2,4     ->    2
 6 -> 1,2,3,6   ->   23
 9 -> 1,3,9     ->    3
21 -> 1,3,7,21  ->   37
22 -> 1,2,11,22 ->  211
25 -> 1,5,25    ->    5
33 -> 1,3,11,33 ->  311
39 -> 1,3,13,39 ->  313
...
26 is not a member :
=20
26 -> 1,2,13,26 -> 213 -> 1,3,71,213 -> 371 -> 1,7,53,371 -> 753 -> 1,3,251=
,753 -> 3251 is prime
=20
but 753 will be a member.
=20
=20
I like 54:
=20
54 -> 1,2,3,6,9,18,27,54 -> 23691827 prime
=20
Best,
=C9.
=20
-----
=20
(I've used Magma to compute the factorization, plus Edwin Clarkk's advice:
=20
> [Math-Fun]
>
> (...)
> Use for example the command Divisors(12345678); at this site
>
>   http://magma.maths.usyd.edu.au/calc/
>
> and click on evaluate. You will get
>
> [ 1, 2, 3, 6, 9, 18, 47, 94, 141, 282, 423, 846, 14593, 29186, 43779,
> 87558, 131337, 262674, 685871, 1371742, 2057613, 4115226, 6172839,
> 12345678 ]
>
> Total time: 0.350 seconds, Total memory usage: 6.46MB


=20
=20
=20
Please do not send any message during this period as you will
not be able to check whether it has been received or not by
the mailing list robot.  Anyway, they would be distributed
only after the servers are rebooted.

I will send a mail to the list so that everyone knows operations
are back to normal.  Thanks in advance for your patience and
your discipline in that matter.

with my best regards,

Olivier GERARD

PS: As usual, any inquiries about this matter should be directed
to me (olivier.gerard at gmail.com or olivier.gerard at paris7.jussieu.fr)
and not to the list.

=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D

Chers abonn=E9s de la liste seqfan

Arr=EAt temporaire du serveur h=E9bergeant la liste

JEUDI 12 Juillet 18h, heure de Paris, pour les fran=E7ais,

jusqu'au

VENDREDI 13 Juillet au matin (probablement 9h)


S'il vous plait, n'envoyez pas de message pendant cette p=E9riode,
vous ne pourriez pas savoir si il a =E9t=E9 bien re=E7u, et vous risqueriez
de provoquer une confusion ou des doublons.
J'enverrai un courier =E0 la liste quand la situation sera revenue
=E0 la normale.

Merci d'avance pour votre patience et votre discipline.


Cordialement,

Olivier GERARD


PS: Comme d'habitude, si vous avez des questions =E0 ce sujet, =E9crivez
moi directement (olivier.gerard at gmail.com ou
olivier.gerard at paris7.jussieu.fr ) et pas =E0 la liste.





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<BODY>
<DIV id=3DidOWAReplyText34525 dir=3Dltr>
<DIV dir=3Dltr><FONT face=3D"Courier New" color=3D#000000 size=3D2>Hello=20
SeqFans,</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>could someone check and =
compute a=20
hundred terms or so of the seq below?</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New"=20
size=3D2>4,6,9,21,22,25,33,39,46,49,51,54,...</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>Integers n&nbsp;whose "o=
rdered=20
concatenation" of all divisors (except 1 and n) is a prime.</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&nbsp;n&nbsp;&nbsp;&nbsp=
;&nbsp;=20
div.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;prime</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&nbsp;4 -&gt;=20
1,2,4&nbsp;&nbsp;&nbsp;&nbsp; -&gt; &nbsp; &nbsp;2</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&nbsp;6 -&gt; 1,2,3,6&nb=
sp;&nbsp;=20
-&gt;&nbsp;&nbsp; 23</FONT></DIV></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&nbsp;9 -&gt; 1,3,9&nbsp=
;&nbsp;=20
&nbsp; -&gt;&nbsp;&nbsp;&nbsp; 3</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>21 -&gt; 1,3,7,21&nbsp;=
=20
-&gt;&nbsp;&nbsp; 37</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>22 -&gt; 1,2,11,22 -&gt;=
&nbsp;=20
211</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>25 -&gt; 1,5,25&nbsp;&nb=
sp;&nbsp;=20
-&gt;&nbsp;&nbsp;&nbsp; 5</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>33 -&gt; 1,3,11,33 -&gt;=
&nbsp;=20
311</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>39 -&gt; 1,3,13,39 -&gt;=
&nbsp;=20
313</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>...</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>26 is not a member :</FO=
NT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>26 -&gt; 1,2,13,26 -&gt;=
 213 -&gt;=20
1,3,71,213 -&gt; 371 -&gt; 1,7,53,371 -&gt; 753 -&gt; 1,3,251,753 -&gt; 325=
1 is=20
prime</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>but 753 will be a=20
member.</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>I like 54:</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>54 -&gt; 1,2,3,6,9,18,27=
,54 -&gt;=20
23691827 prime</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>Best,</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>=C9.</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>-----</FONT></DIV>
<DIV dir=3Dltr>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>(I've used Magma to comp=
ute the=20
factorization, plus Edwin Clarkk's advice:</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&gt; [Math-Fun]</FONT></=
DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&gt;</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&gt; (...)</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&gt; Use for example the=
 command=20
Divisors(12345678); at this site<BR>&gt;<BR>&gt;&nbsp;&nbsp; <A=20
href=3D"/exchweb/bin/redir.asp?URL=3Dhttp://magma.maths.usyd.edu.au/calc/"=
=20
target=3D_blank>http://magma.maths.usyd.edu.au/calc/</A><BR>&gt;<BR>&gt; an=
d click=20
on evaluate. You will get<BR>&gt;<BR>&gt; [ 1, 2, 3, 6, 9, 18, 47, 94, 141,=
 282,=20
423, 846, 14593, 29186, 43779,<BR>&gt; 87558, 131337, 262674, 685871, 13717=
42,=20
2057613, 4115226, 6172839,<BR>&gt; 12345678 ]<BR>&gt;<BR>&gt; Total time: 0=
.350=20
seconds, Total memory usage: 6.46MB<BR><BR></FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT><FONT face=3D"Courier New" size=3D2>Please do not send=
 any=20
message during this period as you will<BR>not be able to check whether it h=
as=20
been received or not by<BR>the mailing list robot.&nbsp; Anyway, they would=
 be=20
distributed<BR>only after the servers are rebooted.<BR><BR>I will send a ma=
il to=20
the list so that everyone knows operations<BR>are back to normal.&nbsp; Tha=
nks=20
in advance for your patience and<BR>your discipline in that matter.<BR><BR>=
with=20
my best regards,<BR><BR>Olivier GERARD<BR><BR>PS: As usual, any inquiries a=
bout=20
this matter should be directed<BR>to me (olivier.gerard at gmail.com or=20
olivier.gerard at paris7.jussieu.fr)<BR>and not to the=20
list.<BR><BR>=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D<BR>=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D<BR><BR>Chers=20
abonn=E9s de la liste seqfan<BR><BR>Arr=EAt temporaire du serveur h=E9berge=
ant la=20
liste<BR><BR>JEUDI 12 Juillet 18h, heure de Paris, pour les=20
fran=E7ais,<BR><BR>jusqu'au<BR><BR>VENDREDI 13 Juillet au matin (probableme=
nt=20
9h)<BR><BR><BR>S'il vous plait, n'envoyez pas de message pendant cette=20
p=E9riode,<BR>vous ne pourriez pas savoir si il a =E9t=E9 bien re=E7u, et v=
ous=20
risqueriez<BR>de provoquer une confusion ou des doublons.<BR>J'enverrai un=
=20
courier =E0 la liste quand la situation sera revenue<BR>=E0 la normale.<BR>=
<BR>Merci=20
d'avance pour votre patience et votre=20
discipline.<BR><BR><BR>Cordialement,<BR><BR>Olivier GERARD<BR><BR><BR>PS: C=
omme=20
d'habitude, si vous avez des questions =E0 ce sujet, =E9crivez<BR>moi direc=
tement=20
(olivier.gerard at gmail.com ou<BR>olivier.gerard at paris7.jussieu.fr ) et pas =
=E0 la=20
liste.<BR><BR><BR><BR></FONT></DIV></FONT>

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From Eric.Angelini at kntv.be  Thu Jul 19 22:38:44 2007
From: Eric.Angelini at kntv.be (Eric Angelini)
Date: Thu, 19 Jul 2007 22:38:44 +0200
Subject: Divisors concatenated shape a prime
Message-ID: <F05775148D000C4B9321A5113D53CB7D50D20B@KNTV-SERVER01.kntv.local>

Sorry for the bad header -- and the old "tail"... I'm not working on my  "normal" computer and quite confused :-/
Best,
?.

________________________________


Hello SeqFans,
could someone check and compute a hundred terms or so of the seq below?
 
4,6,9,21,22,25,33,39,46,49,51,54,...
 
Integers n whose "ordered concatenation" of all divisors (except 1 and n) is a prime.
 
 n     div.         prime
 4 -> 1,2,4     ->    2
 6 -> 1,2,3,6   ->   23
 9 -> 1,3,9     ->    3
21 -> 1,3,7,21  ->   37
22 -> 1,2,11,22 ->  211
25 -> 1,5,25    ->    5
33 -> 1,3,11,33 ->  311
39 -> 1,3,13,39 ->  313
...
26 is not a member :
 
26 -> 1,2,13,26 -> 213 -> 1,3,71,213 -> 371 -> 1,7,53,371 -> 753 -> 1,3,251,753 -> 3251 is prime
 
but 753 will be a member.
 
 
I like 54:
 
54 -> 1,2,3,6,9,18,27,54 -> 23691827 prime
 
Best,
?.
 
-----
 
(I've used Magma to compute the factorization, plus Edwin Clarkk's advice:
 
> [Math-Fun]
>
> (...)
> Use for example the command Divisors(12345678); at this site
>
>   http://magma.maths.usyd.edu.au/calc/
>
> and click on evaluate. You will get
>
> [ 1, 2, 3, 6, 9, 18, 47, 94, 141, 282, 423, 846, 14593, 29186, 43779,
> 87558, 131337, 262674, 685871, 1371742, 2057613, 4115226, 6172839,
> 12345678 ]
>
> Total time: 0.350 seconds, Total memory usage: 6.46MB


 
 
-------------- next part --------------
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From simon.plouffe at gmail.com  Thu Jul 19 22:54:57 2007
From: simon.plouffe at gmail.com (Simon Plouffe)
Date: Thu, 19 Jul 2007 16:54:57 -0400
Subject: Divisors concatenated shape a prime
In-Reply-To: <F05775148D000C4B9321A5113D53CB7D50D20B@KNTV-SERVER01.kntv.local>
References: <F05775148D000C4B9321A5113D53CB7D50D20B@KNTV-SERVER01.kntv.local>
Message-ID: <33a322bc0707191354h70440f5amb6693976ff15c9a5@mail.gmail.com>

hello I made
this simple maple routine :
###############################
with(numtheory):

for k from 5 to 1e99 do:
v0:=divisors(k):

nn:=nops(v0):
if nn > 3 then
v1:=[seq(v0[j],j=2..nn-1)]:
v2:=cat(seq(convert(v1[n],string),n=1..nops(v1))):
v3:=parse(v2):
if isprime(v3) = true then lprint(k,v3) fi:
fi:
od:
#################################

and here is some output from it
(see the attachement)
#################################

Simon plouffe
-------------- next part --------------
6, 23
21, 37
22, 211
33, 311
39, 313
46, 223
51, 317
54, 23691827
58, 229
78, 236132639
82, 241
93, 331
99, 391133
111, 337
115, 523
133, 719
141, 347
142, 271
147, 372149
153, 391751
154, 2711142277
159, 353
162, 236918275481
166, 283
174, 236295887
177, 359
186, 236316293
187, 1117
189, 379212763
201, 367
205, 541
219, 373
226, 2113
235, 547
237, 379
247, 1319
249, 383
253, 1123
262, 2131
267, 389
274, 2137
286, 211132226143
291, 397
294, 23671421424998147
301, 743
318, 23653106159
319, 1129
327, 3109
333, 3937111
355, 571
358, 2179
366, 23661122183
387, 3943129
391, 1723
402, 23667134201
411, 3137
427, 761
459, 39172751153
478, 2239
489, 3163
501, 3167
502, 2251
505, 5101
511, 773
531, 3959177
535, 5107
538, 2269
543, 3181
562, 2281
565, 5113
573, 3191
583, 1153
586, 2293
589, 1931
598, 213232646299
606, 236101202303
622, 2311
639, 3971213
654, 236109218327
657, 3973219
658, 27144794329
679, 797
687, 3229
694, 2347
697, 1741
711, 3979237
721, 7103
726, 23611223366121242363
747, 3983249
753, 3251
759, 311233369253
763, 7109
766, 2383
771, 3257
778, 2389
781, 1171
793, 1361
799, 1747
801, 3989267
813, 3271
822, 236137274411
835, 5167
871, 1367
889, 7127
895, 5179
901, 1753
921, 3307
931, 71949133
934, 2467
939, 3313
943, 2341
949, 1373
957, 311293387319
985, 5197
993, 3331
994, 271471142497
1003, 1759
1006, 2503
1017, 39113339
1041, 3347
1042, 2521
1057, 7151
1077, 3359
1078, 27111422497798154539
1081, 2347
1083, 31957361
1114, 2557
1119, 3373
1135, 5227
1143, 39127381
1147, 3137
1165, 5233
1167, 3389
1186, 2593
1234, 2617
1243, 11113
1254, 23611192233385766114209418627
1294, 2647
1299, 3433
1318, 2659
1339, 13103
1341, 39149447
1347, 3449
1351, 7193
1354, 2677
1362, 236227454681
1366, 2683
1371, 3457
1383, 3461
1387, 1973
1389, 3463
1401, 3467
1405, 5281
1411, 1783
1417, 13109
1426, 223314662713
1435, 573541205287
1438, 2719
1441, 11131
1467, 39163489
1473, 3491
1474, 2112267134737
1477, 7211
1497, 3499
1501, 1979
1513, 1789
1518, 23611222333466669138253506759
1537, 2953
1551, 3113347141517
1581, 317315193527
1594, 2797
1603, 7229
1611, 39179537
1615, 517198595323
1623, 3541
1629, 39181543
1633, 2371
1639, 11149
1641, 3547
1645, 573547235329
1651, 13127
1666, 271417344998119238833
1671, 3557
1686, 236281562843
1713, 3571
1714, 2857
1719, 39191573
1735, 5347
1749, 3113353159583
1774, 2887
1779, 3593
1798, 229315862899
1819, 17107
1821, 3607
1839, 3613
1843, 1997
1851, 3617
1855, 573553265371
1893, 3631
1899, 39211633
1902, 236317634951
1903, 11173
1906, 2953
1909, 2383
1929, 3643
1942, 2971
1963, 13151
1977, 3659
1981, 7283
2001, 323296987667
2013, 3113361183671
2019, 3673
2026, 21013
2031, 3677
2034, 2369181132263396781017
2038, 21019
2043, 39227681
2047, 2389
2059, 2971
2062, 21031
2073, 3691
2077, 3167
2094, 2363496981047
2095, 5419
2103, 3701
2119, 13163
2122, 21061
2127, 3709
2149, 7307
2155, 5431
2157, 3719
2167, 11197
2169, 39241723
2173, 4153
2181, 3727
2199, 3733
2214, 236918274154821232463697381107
2215, 5443
2217, 3739
2245, 5449
2257, 3761
2259, 39251753
2283, 3761
2307, 3769
2317, 7331
2326, 21163
2329, 17137
2343, 3113371213781
2374, 21187
2386, 21193
2391, 3797
2395, 5479
2419, 4159
2421, 39269807
2442, 2361122333766741112224078141221
2443, 7349
2463, 3821
2469, 3823
2479, 3767
2514, 2364198381257
2515, 5503
2527, 719133361
2529, 39281843
2554, 21277
2559, 3853
2566, 21283
2586, 2364318621293
2589, 3863
2605, 5521
2629, 11239
2631, 3877
2637, 39293879
2638, 21319
2641, 19139
2643, 3881
2679, 3194757141893
2721, 3907
2733, 3911
2742, 2364579141371
2757, 3919
2761, 11251
2763, 39307921
2773, 4759
2785, 5557
2787, 3929
2815, 5563
2827, 11257
2839, 17167
2841, 3947
2845, 5569
2866, 21433
2883, 33193961
2901, 3967
2914, 2314762941457
2922, 2364879741461
2923, 3779
2929, 29101
2962, 21481
2967, 3234369129989
2974, 21487
2977, 13229
2983, 19157
2986, 21493
2994, 2364999981497
2998, 21499
3013, 23131
3031, 7433
3039, 31013
3046, 21523
3057, 31019
3097, 19163
3099, 31033
3115, 573589445623
3117, 31039
3118, 21559
3123, 393471041
3133, 13241
3139, 4373
3141, 393491047
3153, 31051
3154, 21938831661577
3178, 27142274541589
3189, 31063
3199, 7457
3202, 21601
3205, 5641
3207, 31069
3226, 21613
3235, 5647
3246, 23654110821623
3247, 17191
3265, 5653
3273, 31091
3286, 23153621061643
3295, 5659
3322, 211221513021661
3333, 311331013031111
3346, 27142394781673
3357, 393731119
3369, 31123
3409, 7487
3415, 5683
3417, 31751672011139
3421, 11311
3439, 19181
3451, 71729119203493
3453, 31151
3459, 31153
3487, 11317
3493, 7499
3505, 5701
3526, 2414382861763
3543, 31181
3573, 393971191
3574, 21787
3579, 31193
3589, 3797
3597, 311331093271199
3609, 394011203
3657, 32353691591219
3661, 7523
3669, 31223
3693, 31231
3711, 31237
3715, 5743
3742, 21871
3747, 31249
3777, 31259
3786, 23663112621893
3787, 7541
3789, 394211263
3799, 29131
3829, 7547
3831, 31277
3841, 23167
3883, 11353
3901, 4783
3921, 31307
3957, 31319
3963, 31321
3973, 29137
3979, 23173
3981, 31327
3987, 394431329
3994, 21997
4006, 22003
4009, 19211
4011, 37211915731337
4039, 7577
4047, 31957712131349
4054, 22027
4063, 17239
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139915, 527983
139917, 346639
139927, 178231
139942, 2112263611272269971
139945, 5136521531076527989
139947, 346649
140002, 270001
140005, 528001
140011, 197369
140017, 163859
140019, 3113342431272946673
140026, 2531061321264270013
140041, 1129319439482912731
140083, 711973
140101, 1316982910777
140137, 433259
140161, 720023
140166, 23691318263978117234599119817973594539177871078215574233614672270083
140214, 236233694673870107
140229, 391558146743
140233, 1773113124119218249
140242, 270121
140253, 346751
140257, 1310789
140293, 239587
140298, 2366713420134940269810472094233834676670149
140311, 193727
140314, 270157
140326, 270163
140377, 229613
140439, 3133916927750783136011080346813
140446, 270223
140458, 270229
140482, 270241
140485, 528097
140493, 346831
140539, 7171191181826720077
140542, 270271
140554, 231622267453470277
140571, 391561946857
140601, 346867
140623, 720089
140631, 346877
140653, 433271
140674, 237741901380270337
140707, 720101
140758, 270379
140769, 391564146923
140782, 243861637327470391
140799, 346933
140815, 528163
140823, 391564746941
140851, 831697
140857, 791783
140902, 270451
140919, 3107321439131746973
141019, 314549
141031, 1112821
141093, 396118325754977123131567747031
141094, 219384779941588931501178630023713742670547
141097, 111011271111139712827
141109, 731933
141111, 391567947037
141123, 347041
141139, 532663
141153, 347051
141166, 270583
141171, 347057
141177, 347059
141187, 592393
141189, 319572477743147063
141211, 720173
141238, 270619
141247, 1371031
141289, 236143
141331, 791789
141334, 270667
141367, 373379
141387, 347129
141418, 270709
141429, 347143
141445, 528289
141451, 373823
141454, 2107214661132270727
141463, 749288720209
141466, 2132654411088270733
141483, 347161
141487, 151937
141553, 353401
141583, 1310891
141597, 391573347199
141607, 192925755148837453
141654, 236236094721870827
141682, 270841
141726, 23613232639466978791381582372994745988971027179418172054308136345451616210902236214724270863
141753, 347251
141799, 747329431301720257
141805, 579359395179528361
141826, 270913
141847, 831709
141859, 1271117
141861, 347287
141865, 517851669834528373
141874, 270937
141889, 1112899
141891, 347297
141901, 413461
141934, 213265310310620668913391378267854591091870967
141951, 347317
141985, 573365389194528397
141997, 149953
142015, 528403
142021, 1112911
142071, 323296971872136671633200120594899617747357
142126, 217935839779471063
142165, 528433
142179, 383249571171347393
142203, 3107321443132947401
142219, 7117718471292920317
142227, 391580347409
142234, 219381973613947223743748671117
142255, 5231151237618528451
142258, 271129
142267, 1131259
142279, 791801
142321, 314591
142347, 323692063618947449
142351, 1112941
142353, 391581747451
142363, 1347233611302910951
142386, 2361938571141249249837477494237314746271193
142399, 157907
142402, 2132654771095471201
142411, 532687
142413, 3371111283384947471
142419, 329871637491147473
142443, 37917192149515763119133147153171323357399441833931969107111972261249927932907678374978379158272034947481
142447, 181787
142483, 1112953
142489, 891601
142507, 314597
142513, 720359
142522, 271261
142539, 347513
142549, 1112959
142599, 347533
142606, 2113226631126271303
142633, 197507
142651, 294919
142666, 271333
142689, 347563
142693, 314603
142702, 2714101932038671351
142717, 433319
142726, 271363
142741, 349409
142773, 347591
142777, 672131
142795, 528559
142801, 612341
142813, 1112983
142827, 347609
142831, 1310987
142849, 720407
142858, 271429
142887, 347629
142909, 1310993
142942, 271471
142966, 271483
143029, 281509
143071, 173827
143098, 271549
143151, 347717
143155, 528631
143179, 672137
143194, 271597
143206, 27145310619337138674213512702102292045871603
143217, 391591347739
143253, 3911339914474341130231591747751
143322, 236238874777471661
143337, 347779
143349, 371213673201947783
143358, 236238934778671679
143383, 1271129
143386, 271693
143397, 39274711314133942310171269305153111593347799
143473, 1113043
143502, 236239174783471751
143523, 3937111333431129338791594747841
143581, 672143
143598, 23671314212639427891182263273526546789157818413419368255236838102571104620514239334786671799
143599, 178447
143601, 315131745395147867
143611, 1311047
143614, 271807
143631, 391595947877
143662, 2109218659131871831
143673, 383249577173147891
143698, 271849
143707, 1311097
143722, 271861
143734, 271867
143739, 391597147913
143746, 241821753350671873
143757, 391597347919
143758, 271879
143761, 233617
143799, 347933
143851, 971483
143853, 347951
143886, 236239814796271943
143887, 197573
143907, 347969
143929, 163883
143943, 347981
143973, 391751153941282384691599747991
144051, 348017
144097, 1031399
144106, 272053
144118, 2132326462412994825983133554362661108672059
144121, 167863
144133, 1113103
144177, 3111733511872575617712827436984811310748059
144181, 314651
144262, 217344243848672131
144277, 720611
144297, 391603348099
144429, 331931553465948143
144454, 272227
144489, 348163
144493, 1311103
144499, 229631
144517, 178501
144519, 367201719215748173
144523, 433361
144538, 272269
144558, 236918275426775354803116062240934818672279
144574, 272287
144601, 236287
144607, 413527
144631, 612371
144663, 348221
144679, 149971
144682, 272341
144703, 1311131
144706, 272353
144717, 348239
144721, 178513
144741, 348247
144766, 272383
144777, 348259
144783, 391608748261
144799, 197621
144841, 241601
144853, 413533
144858, 2367142142344968981034720694241434828672429
144865, 573541392069528973
144943, 193751
144997, 612377
145039, 433373
145053, 397121322763968120431611748351
145057, 1113187
145059, 348353
145066, 272533
145101, 3113343971319148367
145129, 178537
145131, 372169112073348377
145135, 529027
145147, 173839
145165, 529033
145183, 473089
145186, 222931745863472593
145201, 720743
145221, 348407
145249, 1311173
145269, 391614148423
145279, 1311109
145311, 348437
145357, 1371061
145363, 612383
145389, 348463
145401, 317512851855348467
145411, 720773
145426, 219384386891788171634169133823827765472713
145429, 236323
145438, 272719
145461, 348487
145498, 223463163632672749
145542, 2361271912543813825737621146242574851472771
145567, 236329
145581, 348527
145582, 283166877175472791
145645, 529129
145653, 3471411033309948551
145683, 391618748561
145689, 348563
145699, 367397
145713, 348571
145714, 241821777355472857
145747, 747329443310120821
145761, 3711213377231631189344176941132512082348587
145765, 529153
145774, 223463169633872887
145795, 5136522431121529159
145798, 226927153854272899
145801, 211691
145809, 391751153953285985771620148603
145822, 272911
145837, 413557
145843, 172337339163418579
145891, 373943
145918, 272959
145927, 731999
145983, 348661
145999, 720857
146007, 391622348669
146017, 151967
146043, 392781243601180354091622748681
146049, 389267547164148683
146061, 391622948687
146086, 273043
146089, 1391051
146101, 193757
146107, 1311239
146122, 273061
146131, 295039
146149, 178597
146193, 348731
146209, 720887
146254, 273127
146263, 1311251
146271, 348757
146278, 211226110912221867111991342239866491329873139
146281, 197699
146293, 720899
146311, 1147283517311313301
146331, 397121322963968720611625948777
146337, 348779
146346, 236243914878273173
146353, 178609
146362, 273181
146374, 216332644989873187
146379, 359177827248148793
146398, 2714104572091473199
146413, 314723
146446, 237741979395873223
146482, 2714104632092673241
146517, 372169772093148839
146526, 236244214884273263
146545, 573553792653713955531855276541872093529309
146569, 1031423
146629, 720947
146635, 529327
146649, 348883
146659, 178627
146667, 348889
146671, 723161911637720953
146707, 1113337
146731, 1311287
146778, 2361734511021439287843178634244634892673389
146863, 175316390127718639
146866, 273433
146874, 23671314212639427891182269273538546807161418833497376656496994104911129820982244794895873437
146881, 720983
146911, 1071373
146959, 179821
146967, 348989
146973, 348991
146994, 236244994899873497
147034, 273517
147043, 1311311
147055, 529411
147061, 199739
147079, 197741
147085, 5231151279639529417
147091, 721013
147115, 529423
147129, 349043
147202, 2112266911338273601
147226, 273613
147243, 349081
147286, 273643
147307, 197753
147327, 349109
147349, 295081
147363, 349121
147418, 273709
147421, 197759
147454, 273727
147462, 2367142142351170221053321066245774915473731
147513, 349171
147531, 349177
147577, 178681
147589, 1311353
147597, 349199
147621, 349207
147622, 231622381476273811
147631, 1113421
147633, 349211
147643, 191773
147655, 529531
147681, 396118326954980724211640949227
147691, 1131307
147741, 311333711112136340712211331399344771343149247
147781, 178693
147817, 532789
147838, 219338338676673919
147886, 273943
147894, 236157314471942246494929873947
147895, 5115526891344529579
147961, 1113451
147973, 721139
147979, 1311383
147985, 517851741870529597
147993, 349331
148033, 179827
148107, 349369
148117, 732029
148126, 2112267331346674063
148129, 167887
148141, 721163
148183, 721169
148186, 274093
148203, 3911273399297499149744915489134731646749401
148213, 1316987711401
148219, 192926955151117801
148227, 349409
148231, 227653
148233, 349411
148263, 373219677203149421
148278, 2361326397819013802570311406247134942674139
148281, 372123691613074839212149644770612118349427
148327, 236449
148354, 274177
148369, 131011131313146911413
148402, 274201
148431, 349477
148453, 532801
148474, 2611221217243474237
148497, 349499
148563, 391751153971291387391650749521
148587, 349529
148599, 3911193357799917120923762771186915011881260745037821135091651149533
148615, 529723
148683, 329871709512749561
148686, 236247814956274343
148702, 214929849999874351
148705, 529741
148714, 274357
148722, 2367142142354170821062321246247874957474361
148761, 391652949587
148801, 178753
148803, 319325757977149601
148809, 349603
148849, 473167
148863, 311133339143347429104138174511114511353349621
148882, 274441
148903, 171932346178378759
148906, 274453
148942, 274471
148947, 3131379393113749649
148978, 274489
148981, 721283
148989, 349663
149001, 349667
149041, 1031447
149091, 349697
149095, 529819
149107, 7174911917983312533043877121301
149163, 372171032130949721
149167, 433469
149203, 314813
149218, 274609
149233, 721319
149329, 592531
149353, 233641
149361, 349787
149386, 2113226661132274693
149403, 349801
149437, 295153
149449, 199751
149493, 349831
149527, 741287521364721361
149539, 1311503
149562, 23679141821426312611872374356171228309106831661821366249274985474781
149613, 349871
149635, 529927
149653, 721379
149662, 274831
149686, 274843
149719, 178807
149722, 274861
149739, 3193757711112137031349210926274047788149913
149757, 349919
149779, 721397
149781, 349927
149785, 5291451033516529957
149817, 349939
149901, 329871723516949967
149902, 224131148262274951
149923, 178819
149946, 2366713420137340274611192238249914998274973
149949, 391666149983
149977, 473191
150099, 350033
150121, 2361107140324616527
150127, 178831
150166, 275083
150238, 2112268291365875119
150253, 971549
150261, 350087
150265, 541205733366530053
150279, 350093
150291, 391669950097
150313, 831811
150321, 389267563168950107
150322, 275161
150333, 350111
150334, 275167
150367, 721481
150421, 359419
150438, 236250735014675219
150454, 275227
150463, 379397
150477, 350159
150493, 721499
150529, 1091381
150538, 275269
150541, 473203
150561, 391672950187
Interrupted

From simon.plouffe at gmail.com  Thu Jul 19 23:10:11 2007
From: simon.plouffe at gmail.com (Simon Plouffe)
Date: Thu, 19 Jul 2007 17:10:11 -0400
Subject: Divisors concatenated shape a prime
In-Reply-To: <F05775148D000C4B9321A5113D53CB7D50D20B@KNTV-SERVER01.kntv.local>
References: <F05775148D000C4B9321A5113D53CB7D50D20B@KNTV-SERVER01.kntv.local>
Message-ID: <33a322bc0707191410x7c82df34qb3b68408bbab7077@mail.gmail.com>

Hello,

And for the number 339066 the prime generated has 255 digits:

here it is :
339066, 2367913141821232627394246546369788191117126138161162182189207234273299322351378414483546567598621702819897966105
311341242144916381794186320932106245726912898372641864347491453826279737180738694125581304114742161461883724219260823767
44843856511113022169533

je continue les calculs.

simon plouffe




From Eric.Angelini at kntv.be  Thu Jul 19 23:19:48 2007
From: Eric.Angelini at kntv.be (Eric Angelini)
Date: Thu, 19 Jul 2007 23:19:48 +0200
Subject: =?iso-8859-1?Q?RE=A0=3A_Divisors_concatenated_shape_a_prime?=
Message-ID: <F05775148D000C4B9321A5113D53CB7D50D20C@KNTV-SERVER01.kntv.local>

 
Geee, Simon !
Simple routine but... great result ! Waow !
Thanks to you and everyone -- this list is a must ;-)
Best,
?.
 
 

________________________________

De: Simon Plouffe [mailto:simon.plouffe at gmail.com]
Date: jeu. 19/07/2007 22:54
?: Eric Angelini
Cc: seqfan at ext.jussieu.fr
Objet : Re: Divisors concatenated shape a prime



hello I made
this simple maple routine :
###############################
with(numtheory):

for k from 5 to 1e99 do:
v0:=divisors(k):

nn:=nops(v0):
if nn > 3 then
v1:=[seq(v0[j],j=2..nn-1)]:
v2:=cat(seq(convert(v1[n],string),n=1..nops(v1))):
v3:=parse(v2):
if isprime(v3) = true then lprint(k,v3) fi:
fi:
od:
#################################

and here is some output from it
(see the attachement)
#################################

Simon plouffe


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From njas at research.att.com  Fri Jul 20 00:27:53 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Thu, 19 Jul 2007 18:27:53 -0400 (EDT)
Subject: Divisors concatenated shape a prime
Message-ID: <200707192227.l6JMRr4e351446@fry.research.att.com>

sequence A037274, of course?  Very similar to yours,
Return-Path: <maxale at gmail.com>
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Message-ID: <d3dac270707191613u57980344yef08bacd01a44fa3 at mail.gmail.com>
Date: Thu, 19 Jul 2007 16:13:08 -0700
From: "Max Alekseyev" <maxale at gmail.com>
To: njas at research.att.com
Subject: Re: Divisors concatenated shape a prime
Cc: seqfan at ext.jussieu.fr, "Eric Angelini" <Eric.Angelini at kntv.be>
In-Reply-To: <200707192227.l6JMRr4e351446 at fry.research.att.com>
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Neil,

It seems that A037274 (and related sequences, e.g., A037275) is
missing the "base" keyword.

Regards,
Max

On 7/19/07, N. J. A. Sloane <njas at research.att.com> wrote:
> Eric,  you know about the "home primes"
> sequence A037274, of course?  Very similar to yours,
> and known to be hard!
> Neil
>




From warut822 at gmail.com  Fri Jul 20 04:32:50 2007
From: warut822 at gmail.com (Warut Roonguthai)
Date: Fri, 20 Jul 2007 09:32:50 +0700
Subject: Lengths Of Runs In The #-Of-Divisors Sequence
In-Reply-To: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
References: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
Message-ID: <482644420707191932s56b4979p6fc0c36fd3576d31@mail.gmail.com>

On 7/17/07, Leroy Quet <qq-quet at mindspring.com> wrote:
>
> It seems VERY likely to me that there is no infinite string of 1's, or of
> anything else, in sequence A131789 (ie. the terms of A131790 are all
> finite).
>
> Can it be PROVED that all terms of A131790 are finite, possibly using
> Hardy and Wright or some other such reference?

Since there are infinitely many primes, 1 will appear infinitely often
in A131789. But there cannot be an infinite string of 1's in A131789
because there exist infinitely many m such that d(m) = d(m+1) as
proved by Roger Heath-Brown in 1984. So all terms of A131790 are
finite.

Warut



Simon Plouffe:

> And for the number 339066 the prime generated has 255 digits

Looking at just the record values, from here on...

{record-index, seed-number, digits-in-target-prime}

{41,339066,255}
{42,594594,317}
{43,902538,328}
{44,1750014,341}
{45,2254098,346}
{46,3174138,352}
{47,3467646,354}
{48,3818178,446}
{49,3913434,447}
{50,8795358,501}
{51,9489018,502}
{52,9522414,503}
{53,13891878,511}
{54,14139762,514}
{55,14167494,515}
{56,19803966,522}
{57,23978262,529}
{58,24478146,594}
{59,28289898,673}
{60,35837802,1025}




I'm adding Eric's two new sequences: they will
be A120712 and A120713.  I used Simon Plouffe's 
nice Maple program to generate them, but modified
it to give exactly the same terms as Eric had.

To get an analogue of the home primes, A037274, I suppose
we could iterate the map (k -> concatenation of proper
divisors of k) until we reach a prime; then a(n) would
be the prime we eventually reach when starting with n,
or -1 if we never reach a prime.  But what about a(p)
where p is prime?  I guess we set that to equal p.

And we can set a(1) = 1.  So the sequence starts
1 2 3 2 5 23 7 ...
What is a(8)?
I don't know, but I will put it in the OEIS as A120716.
Hopefully someone will extend it.

Neil




Seqfans,  for the new sequence A120716 that I mentioned,
the analogue of home primes, the big question is,
what is a(8)?
we have 

and the divisors of that last number aree

1, 2, 3, 6, 37, 74, 111, 113, 222, 226, 339, 678, 4181, 8362, 12543, 25086,
3192525397, 6385050794, 9577576191, 19155152382, 118123439689, 236246879378,
354370319067, 360755369861, 708740638134, 721510739722, 1082266109583,
2164532219166, 13347948684857, 26695897369714, 40043846054571, 80087692109142,
30132785470246166539693, 60265570940492333079386, 90398356410738499619079,
180796712821476999238158, 1114913062399108161968641, 2229826124798216323937282,
3344739187197324485905923, 3405004758137816818985309, 6689478374394648971811846,
6810009516275633637970618, 10215014274413450456955927,
20430028548826900913911854, 125985176051099222302456433,
251970352102198444604912866, 377955528153297666907369299,
755911056306595333814738598, 96199682896113474519861511083121,
192399365792226949039723022166242, 288599048688340423559584533249363,
577198097376680847119169066498726, 3559388267156198557234875910075477,
7118776534312397114469751820150954, 10678164801468595671704627730226431,
10870564167260822620744350752392673, 21356329602937191343409255460452862,
21741128334521645241488701504785346, 32611692501782467862233052257178019,
65223385003564935724466104514356038, 402210874188650436967540977838528901,
804421748377300873935081955677057802, 1206632622565951310902622933515586703,
2413265245131902621805245867031173406 ]

If we throw away the first and last terms and concatenate the rest,
is that a prime?  Does someone have access to a really good
prime tester?!

Neil

PS I did not check these calculations and it is 05:00




From pxp at rogers.com  Fri Jul 20 06:17:12 2007
From: pxp at rogers.com (Hans Havermann)
Date: Fri, 20 Jul 2007 00:17:12 -0400
Subject: Divisors concatenated shape a prime
In-Reply-To: <33a322bc0707191410x7c82df34qb3b68408bbab7077@mail.gmail.com>
References: <F05775148D000C4B9321A5113D53CB7D50D20B@KNTV-SERVER01.kntv.local> <33a322bc0707191410x7c82df34qb3b68408bbab7077@mail.gmail.com>
Message-ID: <EA720807-1D72-4D3C-ABA3-05D61696B393@rogers.com>

On 7/20/07, N. J. A. Sloane <njas at research.att.com> wrote:

> If we throw away the first and last terms and concatenate the rest,
> is that a prime?

No, it is not prime.
And it is quite a large number that may be hard to factor. I will let
the PARI/GP factor() function run overnight...

Max




From simon.plouffe at gmail.com  Fri Jul 20 13:43:16 2007
From: simon.plouffe at gmail.com (Simon Plouffe)
Date: Fri, 20 Jul 2007 07:43:16 -0400
Subject: ps Re Divisors concatenated shape a prime
In-Reply-To: <d3dac270707200436x7996f230qdeda7d1a243a730e@mail.gmail.com>
References: <200707201002.l6KA2ZKE432256@fry.research.att.com>
	 <d3dac270707200436x7996f230qdeda7d1a243a730e@mail.gmail.com>
Message-ID: <33a322bc0707200443l833a591gc4abfcff9e172a43@mail.gmail.com>

The number is
23637741111132222263396784181836212543250863192525397638\
    5050794957757619119155152382118123439689236246879378\
    3543703190673607553698617087406381347215107397221082\
    2661095832164532219166133479486848572669589736971440\
    0438460545718008769210914230132785470246166539693602\
    6557094049233307938690398356410738499619079180796712\
    8214769992381581114913062399108161968641222982612479\
    8216323937282334473918719732448590592334050047581378\
    1681898530966894783743946489718118466810009516275633\
    6379706181021501427441345045695592720430028548826900\
    9139118541259851760510992223024564332519703521021984\
    4460491286637795552815329766690736929975591105630659\
    5333814738598961996828961134745198615110831211923993\
    6579222694903972302216624228859904868834042355958453\
    3249363577198097376680847119169066498726355938826715\
    6198557234875910075477711877653431239711446975182015\
    0954106781648014685956717046277302264311087056416726\
    0822620744350752392673213563296029371913434092554604\
    5286221741128334521645241488701504785346326116925017\
    8246786223305225717801965223385003564935724466104514\
    3560384022108741886504369675409778385289018044217483\
    7730087393508195567705780212066326225659513109026229\
    33515586703

and divisible by 13 and 47 if I am not mistaking.

the rest (%/13/47) is not prime either but I can't determine
its factors.

simon plouffe




From jvospost3 at gmail.com  Sat Jul 21 00:35:58 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Fri, 20 Jul 2007 15:35:58 -0700
Subject: Concatenated anti-divisors
Message-ID: <5542af940707201535q1a3f22f5xfd0c2bdb892fab6d@mail.gmail.com>

Non-divisor: a number k which does not divide a given
number x. Anti-divisor: a non-divisor k of x with the
property that k is an odd divisor of 2x-1 or 2x+1, or
an even divisor of 2x.

There are no anti-divisors of 1 and 2.

See A066272 for an equivalent definition and also the
number of terms in each row.

Now, if we concatenate antidivisors of integers n>2, we have an
anti-divisor analogue of A106708.

This should be easy for someone to Mathematica-ize and extend, and
leads to the same kind of fun questions as its origin does.

===========
n  a(n)  factorization
3  2     prime
4  3     prime
5  23    prime
6  4     2^2
7  235   5 * 47
8  35    5 * 7
9  26    2 * 13
10 347   prime
11 237   3 * 79
12 58    2 * 29
13 2359  7 * 337
14 349   prime
15 2610  2 * 3^2 * 5 * 29
16 311   prime
17 235711  7 * 151 * 223
18 45712 2^4 * 2857
19 2313  3^2 * 257
20 3813  3 * 31 * 41
===========

--- The On-Line Encyclopedia of Integer Sequences
<oeis at research.att.com> wrote:

 The following is a copy of the email message that
 was sent to njas
  Subject: NEW SEQUENCE FROM Jonathan Vos Post

 %I A000001
 %S A000001 2, 3, 23, 4, 235, 35, 26, 347, 237, 58,
 2359, 349, 2610, 311, 235711, 45712, 2313, 3813
 %N A000001 Replace n by the concatenation of its
 antidivisors.
 %C A000001 Number of antidivisors concatened to form
 a(n) is A066272(n). We may consider prime values of
 the concatenated antidivisor sequence, and we may
 iterate it, i.e. n, a(n), a(a(n)), a(a(a(n))) which
 leads to questions of trajectory, cycles, fixed
 points.
 %e A000001 Anti-divisors of 3 through 20:
 3: 2, so a(3) = 2.
 4: 3, so a(4) = 3.
 5: 2, 3, so a(5) = 23.
 6: 4, so a(6) = 4.
 7: 2, 3, 5, so a(7) = 235.
 17: 2, 3, 5, 7, 11, so a(17) = 235711
 %Y A000001 Cf. A037278, A066272, A120712, A106708,
 A130799.
 %O A000001 3
 %K A000001 ,base,easy,more,nonn,
 %A A000001 Jonathan Vos Post (jvospost2 at yahoo.com),
 Jul 20 2007
 RH
 RA 192.20.225.32




From maximilian.hasler at gmail.com  Sat Jul 21 02:38:38 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Fri, 20 Jul 2007 20:38:38 -0400
Subject: Concatenated anti-divisors
In-Reply-To: <5542af940707201535q1a3f22f5xfd0c2bdb892fab6d@mail.gmail.com>
References: <5542af940707201535q1a3f22f5xfd0c2bdb892fab6d@mail.gmail.com>
Message-ID: <3c3af2330707201738j2685797cr94e6ce49671e21c2@mail.gmail.com>

It appears to me that the definition

%C A066272 If an odd number d in the range 1 < d < n divides N where N
is any one of 2n-1, 2n or 2n+1 then N/d is called an anti-divisor of
n.

is not equivalent to your definition:

> Non-divisor: a number k which does not divide a given
> number x. Anti-divisor: a non-divisor k of x with the
> property that k is an odd divisor of 2x-1 or 2x+1, or
> an even divisor of 2x.

According to the first definition, n=3 cannot have an anti-divisor
since there is no odd d, 1<d<n.

M.H.




From maximilian.hasler at gmail.com  Sat Jul 21 03:35:14 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Fri, 20 Jul 2007 21:35:14 -0400
Subject: Concatenated anti-divisors
In-Reply-To: <3c3af2330707201738j2685797cr94e6ce49671e21c2@mail.gmail.com>
References: <5542af940707201535q1a3f22f5xfd0c2bdb892fab6d@mail.gmail.com>
	 <3c3af2330707201738j2685797cr94e6ce49671e21c2@mail.gmail.com>
Message-ID: <3c3af2330707201835n3b6b37c8r1d7594ea8514a132@mail.gmail.com>

It first appeared as if "your" definition (the second below) could be
found at least on some other web pages. Now I have big doubts. Is
there any *serious* reference for this notion ?
According to "your" definition,
antidiv( 3 ) = { 2, 5, 6, 7 }
since none of these numbers divide 3 and they are either even divisors
of 6 or odd divisors of 5 or 7.

Also, concerning the sequence :

A130799 Triangle read by rows in which row n (n>=3) list the anti-divisors of n.

I cannot see any triangular structure (cf.
http://www.research.att.com/~njas/sequences/table?a=130799
) and AFAICS the only way to find out where the anti-divisors of a
number n will start is to spot violations of strict increasing
monotony
(which is of course not failsafe,  e.g. if a number >2 would have no
antidivisors, or some number would have its largest antidiv. smaller
than the least antidiv of its successor....)

IMHO all these sequences need much clarification and cleanup of definitions.
Insofar more as the "authorative" definition seems to be on an
unavailable web page (and the version I can found via the "wayback"
machine does not contain a clear definition either - again several
"Or, to put it another way..." etc). All in all there is a heavy smell
of *very* original research in the air around this.

M.H.

On 7/20/07, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:
> It appears to me that the definition
>
> %C A066272 If an odd number d in the range 1 < d < n divides N where N
> is any one of 2n-1, 2n or 2n+1 then N/d is called an anti-divisor of
> n.
>
> is not equivalent to your definition:
>
> > Non-divisor: a number k which does not divide a given
> > number x. Anti-divisor: a non-divisor k of x with the
> > property that k is an odd divisor of 2x-1 or 2x+1, or
> > an even divisor of 2x.
>
> According to the first definition, n=3 cannot have an anti-divisor
> since there is no odd d, 1<d<n.
>
> M.H.
>




From maxale at gmail.com  Sat Jul 21 04:19:05 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Fri, 20 Jul 2007 19:19:05 -0700
Subject: Concatenated anti-divisors
In-Reply-To: <3c3af2330707201738j2685797cr94e6ce49671e21c2@mail.gmail.com>
References: <5542af940707201535q1a3f22f5xfd0c2bdb892fab6d@mail.gmail.com>
	 <3c3af2330707201738j2685797cr94e6ce49671e21c2@mail.gmail.com>
Message-ID: <d3dac270707201919v199594c7gab0cb2c18cee2340@mail.gmail.com>

There are glitches in both Jonathan's definition of anti-divisor and
the quoted comment to A066272.

Namely, Jonathan forgot to mention that k < x, and in the comment to
A066272 it should be "1 < d <= n" not "1 < d < n".

After such corrections these two definitions become equivalent and
consistent with the numerical values of A066272.

Max

On 7/20/07, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:
> It appears to me that the definition
>
> %C A066272 If an odd number d in the range 1 < d < n divides N where N
> is any one of 2n-1, 2n or 2n+1 then N/d is called an anti-divisor of
> n.
>
> is not equivalent to your definition:
>
> > Non-divisor: a number k which does not divide a given
> > number x. Anti-divisor: a non-divisor k of x with the
> > property that k is an odd divisor of 2x-1 or 2x+1, or
> > an even divisor of 2x.
>
> According to the first definition, n=3 cannot have an anti-divisor
> since there is no odd d, 1<d<n.




Dear seqfans, There are currently two versions
of the definition of anti-divisor in the OEIS:

%C A066272 If an odd number d in the range 1 < d < n divides N
where N is any one of 2n-1, 2n or 2n+1
then N/d is called an anti-divisor of n.

%e A066272 For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors > 1 {5,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 4.

But this definition fails for n = 3, as someone mentioned last night.
We know from the OEIS that 3 has a single antidivisor, 2. 
According to this definition 3 has no antidivisors.

There is also this program, which I have not checked:
%t A066272 antid[ n_ ] := Select[ Union[ Join[ Select[ Divisors[ 2n - 1 ], OddQ[ # ] && # != 1 & ], Select[ Divisors[ 2n + 1 ], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2*n ], OddQ[ # ] && # != 1 & ] ] ] }, # < n & ] ]; Table[ Length[ antid[ n ] ], {n, 1, 100} ]

The other definition is:

%I A130799
%S A130799 2,3,2,3,4,2,3,5,3,5,2,6,3,4,7,2,3,7,5,8,2,3,5,9,3,4,9,2,6,10,3,11,
%T A130799 2,3,5,7,11,4,5,7,12,2,3,13,3,8,13,2,6,14,3,4,5,9,15,2,3,5,9,15,7,
%U A130799 16,2,3,7,10,17,3,4,17,2,5,6,11,18,3,5,8,11,19,2,3,19,4,12,20,2,3,7
%N A130799 Triangle read by rows in which row n (n>=3) list the anti-divisors of n.
%C A130799 Non-divisor: a number k which does not divide a given number x. Anti-divisor: a non-divisor k of x with the property that k is an odd divisor of 2x-1 or 2x+1, or an even divisor of 2x.
%C A130799 There are no anti-divisors of 1 and 2.
%e A130799 Anti-divisors of 3 through 20:
%e A130799 3: 2
%e A130799 4: 3
%e A130799 5: 2, 3
%e A130799 6: 4
%e A130799 7: 2, 3, 5
%e A130799 8: 3, 5
%e A130799 9: 2, 6

This definition also fails for n = 3: it gives 5 antidivisors,
2,4,5,6,7.

The term anti-divisor seems to be due to Jon Perry.  
I wish I understood the motivation for the definition!

There are links to various webpages of his, but they are all broken
and he has not responded to my emails.

It seems to me that both of the above definitions are incorrect, and

where N is any one of 2n-1, 2n or 2n+1
then d = N/i is called an anti-divisor of n.

Equivalently, an anti-divisor of n is a number d in the range [1..n]
which does not divide n and is either an odd divisor of 2n-1 or 2n+1,
or an even divisor of 2n.

Now both definitions seem to work correctly for n=3, giving
a single anti-divisor, 2.

But I'm not too confident about all this - comments anyone?
As I said, I wish I understood the motivation for the definition!

Neil





From njas at research.att.com  Sat Jul 21 13:41:09 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Sat, 21 Jul 2007 07:41:09 -0400 (EDT)
Subject: definition of anti-divisor
Message-ID: <200707211141.l6LBf9dn795643@fry.research.att.com>

should be changed to:
Definition: If an odd number i in the range 1 < i <= n divides N
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Subject: Re: definition of anti-divisor
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Dr. Sloane,

As far as the second definition, and

> %C A130799 Non-divisor: a number k which does not divide a given number x.
Anti-divisor: a non-divisor k of x with the property that k is an odd
divisor of 2x-1 or 2x+1, or an even divisor of 2x.

> This definition also fails for n = 3: it gives 5 antidivisors,
2,4,5,6,7.

According to the second definition, above, an antidivisor must first be a
non-divisor of x. {4, 5, 6, 7} are not non-divisors of 3. Therefore, I
believe that the second definition is correct as it stands.

Diana

On 7/21/07, N. J. A. Sloane <njas at research.att.com> wrote:
>
>
> Dear seqfans, There are currently two versions
> of the definition of anti-divisor in the OEIS:
>
> %C A066272 If an odd number d in the range 1 < d < n divides N
> where N is any one of 2n-1, 2n or 2n+1
> then N/d is called an anti-divisor of n.
>
> %e A066272 For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd
> divisors > 1 {5,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the
> anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 4.
>
> But this definition fails for n = 3, as someone mentioned last night.
> We know from the OEIS that 3 has a single antidivisor, 2.
> According to this definition 3 has no antidivisors.
>
> There is also this program, which I have not checked:
> %t A066272 antid[ n_ ] := Select[ Union[ Join[ Select[ Divisors[ 2n - 1 ],
> OddQ[ # ] && # != 1 & ], Select[ Divisors[ 2n + 1 ], OddQ[ # ] && # != 1 &
> ], 2n/Select[ Divisors[ 2*n ], OddQ[ # ] && # != 1 & ] ] ] }, # < n & ] ];
> Table[ Length[ antid[ n ] ], {n, 1, 100} ]
>
> The other definition is:
>
> %I A130799
> %S A130799
> 2,3,2,3,4,2,3,5,3,5,2,6,3,4,7,2,3,7,5,8,2,3,5,9,3,4,9,2,6,10,3,11,
> %T A130799
> 2,3,5,7,11,4,5,7,12,2,3,13,3,8,13,2,6,14,3,4,5,9,15,2,3,5,9,15,7,
> %U A130799
> 16,2,3,7,10,17,3,4,17,2,5,6,11,18,3,5,8,11,19,2,3,19,4,12,20,2,3,7
> %N A130799 Triangle read by rows in which row n (n>=3) list the
> anti-divisors of n.
> %C A130799 Non-divisor: a number k which does not divide a given number x.
> Anti-divisor: a non-divisor k of x with the property that k is an odd
> divisor of 2x-1 or 2x+1, or an even divisor of 2x.
> %C A130799 There are no anti-divisors of 1 and 2.
> %e A130799 Anti-divisors of 3 through 20:
> %e A130799 3: 2
> %e A130799 4: 3
> %e A130799 5: 2, 3
> %e A130799 6: 4
> %e A130799 7: 2, 3, 5
> %e A130799 8: 3, 5
> %e A130799 9: 2, 6
>
> This definition also fails for n = 3: it gives 5 antidivisors,
> 2,4,5,6,7.
>
> The term anti-divisor seems to be due to Jon Perry.
> I wish I understood the motivation for the definition!
>
> There are links to various webpages of his, but they are all broken
> and he has not responded to my emails.
>
> It seems to me that both of the above definitions are incorrect, and
> should be changed to:
>
> Definition: If an odd number i in the range 1 < i <= n divides N
> where N is any one of 2n-1, 2n or 2n+1
> then d = N/i is called an anti-divisor of n.
>
> Equivalently, an anti-divisor of n is a number d in the range [1..n]
> which does not divide n and is either an odd divisor of 2n-1 or 2n+1,
> or an even divisor of 2n.
>
> Now both definitions seem to work correctly for n=3, giving
> a single anti-divisor, 2.
>
> But I'm not too confident about all this - comments anyone?
> As I said, I wish I understood the motivation for the definition!
>
> Neil
>
>


-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.

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Content-Transfer-Encoding: 7bit
Content-Disposition: inline

Dr. Sloane,<br><br>As far as the second definition, and <br><br>&gt; %C A130799 Non-divisor: a number k which does not divide a given number
x. Anti-divisor: a non-divisor k of x with the property that k is an
odd divisor of 2x-1 or 2x+1, or an even divisor of 2x.<br><br>&gt; This definition also fails for n = 3: it gives 5 antidivisors,<br>2,4,5,6,7.<br><br>According to the second definition, above, an antidivisor must first be a non-divisor of x. {4, 5, 6, 7} are not non-divisors of 3. Therefore, I believe that the second definition is correct as it stands.
<br><br>Diana<br><br><div><span class="gmail_quote">On 7/21/07, <b class="gmail_sendername">N. J. A. Sloane</b> &lt;<a href="mailto:njas at research.att.com">njas at research.att.com</a>&gt; wrote:</span><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<br>Dear seqfans, There are currently two versions<br>of the definition of anti-divisor in the OEIS:<br><br>%C A066272 If an odd number d in the range 1 &lt; d &lt; n divides N<br>where N is any one of 2n-1, 2n or 2n+1<br>
then N/d is called an anti-divisor of n.<br><br>%e A066272 For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors &gt; 1 {5,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 4.
<br><br>But this definition fails for n = 3, as someone mentioned last night.<br>We know from the OEIS that 3 has a single antidivisor, 2.<br>According to this definition 3 has no antidivisors.<br><br>There is also this program, which I have not checked:
<br>%t A066272 antid[ n_ ] := Select[ Union[ Join[ Select[ Divisors[ 2n - 1 ], OddQ[ # ] &amp;&amp; # != 1 &amp; ], Select[ Divisors[ 2n + 1 ], OddQ[ # ] &amp;&amp; # != 1 &amp; ], 2n/Select[ Divisors[ 2*n ], OddQ[ # ] &amp;&amp; # != 1 &amp; ] ] ] }, # &lt; n &amp; ] ]; Table[ Length[ antid[ n ] ], {n, 1, 100} ]
<br><br>The other definition is:<br><br>%I A130799<br>%S A130799 2,3,2,3,4,2,3,5,3,5,2,6,3,4,7,2,3,7,5,8,2,3,5,9,3,4,9,2,6,10,3,11,<br>%T A130799 2,3,5,7,11,4,5,7,12,2,3,13,3,8,13,2,6,14,3,4,5,9,15,2,3,5,9,15,7,<br>%U A130799 16,2,3,7,10,17,3,4,17,2,5,6,11,18,3,5,8,11,19,2,3,19,4,12,20,2,3,7
<br>%N A130799 Triangle read by rows in which row n (n&gt;=3) list the anti-divisors of n.<br>%C A130799 Non-divisor: a number k which does not divide a given number x. Anti-divisor: a non-divisor k of x with the property that k is an odd divisor of 2x-1 or 2x+1, or an even divisor of 2x.
<br>%C A130799 There are no anti-divisors of 1 and 2.<br>%e A130799 Anti-divisors of 3 through 20:<br>%e A130799 3: 2<br>%e A130799 4: 3<br>%e A130799 5: 2, 3<br>%e A130799 6: 4<br>%e A130799 7: 2, 3, 5<br>%e A130799 8: 3, 5
<br>%e A130799 9: 2, 6<br><br>This definition also fails for n = 3: it gives 5 antidivisors,<br>2,4,5,6,7.<br><br>The term anti-divisor seems to be due to Jon Perry.<br>I wish I understood the motivation for the definition!
<br><br>There are links to various webpages of his, but they are all broken<br>and he has not responded to my emails.<br><br>It seems to me that both of the above definitions are incorrect, and<br>should be changed to:<br>
<br>Definition: If an odd number i in the range 1 &lt; i &lt;= n divides N<br>where N is any one of 2n-1, 2n or 2n+1<br>then d = N/i is called an anti-divisor of n.<br><br>Equivalently, an anti-divisor of n is a number d in the range [1..n]
<br>which does not divide n and is either an odd divisor of 2n-1 or 2n+1,<br>or an even divisor of 2n.<br><br>Now both definitions seem to work correctly for n=3, giving<br>a single anti-divisor, 2.<br><br>But I&#39;m not too confident about all this - comments anyone?
<br>As I said, I wish I understood the motivation for the definition!<br><br>Neil<br><br></blockquote></div><br><br clear="all"><br>-- <br>&quot;God made the integers, all else is the work of man.&quot; <br>L. Kronecker, Jahresber. DMV 2, S. 19.

------=_Part_148534_11559968.1185021085857--




From diana.mecum at gmail.com  Sat Jul 21 14:38:37 2007
From: diana.mecum at gmail.com (Diana Mecum)
Date: Sat, 21 Jul 2007 07:38:37 -0500
Subject: definition of anti-divisor
In-Reply-To: <a851af150707210531s2e01e2b2p3ed2e5c31e635d8d@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <a851af150707210531s2e01e2b2p3ed2e5c31e635d8d@mail.gmail.com>
Message-ID: <a851af150707210538y57a48ce0j32e9a952482038d2@mail.gmail.com>

Dr. Sloane,

As far as the second definition, and

> %C A130799 Non-divisor: a number k which does not divide a given number x.
Anti-divisor: a non-divisor k of x with the property that k is an odd
divisor of 2x-1 or 2x+1, or an even divisor of 2x.

> This definition also fails for n = 3: it gives 5 antidivisors,
2,4,5,6,7.

According to the second definition, above, an antidivisor must first be a
non-divisor of x. {4, 5, 6, 7} are not non-divisors of 3. Therefore, I
believe that the second definition is correct as it stands.

Diana

On 7/21/07, N. J. A. Sloane <njas at research.att.com> wrote:
>
>
> Dear seqfans, There are currently two versions
> of the definition of anti-divisor in the OEIS:
>
> %C A066272 If an odd number d in the range 1 < d < n divides N
> where N is any one of 2n-1, 2n or 2n+1
> then N/d is called an anti-divisor of n.
>
> %e A066272 For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd
> divisors > 1 {5,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the
> anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 4.
>
> But this definition fails for n = 3, as someone mentioned last night.
> We know from the OEIS that 3 has a single antidivisor, 2.
> According to this definition 3 has no antidivisors.
>
> There is also this program, which I have not checked:
> %t A066272 antid[ n_ ] := Select[ Union[ Join[ Select[ Divisors[ 2n - 1 ],
> OddQ[ # ] && # != 1 & ], Select[ Divisors[ 2n + 1 ], OddQ[ # ] && # != 1 &
> ], 2n/Select[ Divisors[ 2*n ], OddQ[ # ] && # != 1 & ] ] ] }, # < n & ] ];
> Table[ Length[ antid[ n ] ], {n, 1, 100} ]
>
> The other definition is:
>
> %I A130799
> %S A130799
> 2,3,2,3,4,2,3,5,3,5,2,6,3,4,7,2,3,7,5,8,2,3,5,9,3,4,9,2,6,10,3,11,
> %T A130799
> 2,3,5,7,11,4,5,7,12,2,3,13,3,8,13,2,6,14,3,4,5,9,15,2,3,5,9,15,7,
> %U A130799
> 16,2,3,7,10,17,3,4,17,2,5,6,11,18,3,5,8,11,19,2,3,19,4,12,20,2,3,7
> %N A130799 Triangle read by rows in which row n (n>=3) list the
> anti-divisors of n.
> %C A130799 Non-divisor: a number k which does not divide a given number x.
> Anti-divisor: a non-divisor k of x with the property that k is an odd
> divisor of 2x-1 or 2x+1, or an even divisor of 2x.
> %C A130799 There are no anti-divisors of 1 and 2.
> %e A130799 Anti-divisors of 3 through 20:
> %e A130799 3: 2
> %e A130799 4: 3
> %e A130799 5: 2, 3
> %e A130799 6: 4
> %e A130799 7: 2, 3, 5
> %e A130799 8: 3, 5
> %e A130799 9: 2, 6
>
> This definition also fails for n = 3: it gives 5 antidivisors,
> 2,4,5,6,7.
>
> The term anti-divisor seems to be due to Jon Perry.
> I wish I understood the motivation for the definition!
>
> There are links to various webpages of his, but they are all broken
> and he has not responded to my emails.
>
> It seems to me that both of the above definitions are incorrect, and
> should be changed to:
>
> Definition: If an odd number i in the range 1 < i <= n divides N
> where N is any one of 2n-1, 2n or 2n+1
> then d = N/i is called an anti-divisor of n.
>
> Equivalently, an anti-divisor of n is a number d in the range [1..n]
> which does not divide n and is either an odd divisor of 2n-1 or 2n+1,
> or an even divisor of 2n.
>
> Now both definitions seem to work correctly for n=3, giving
> a single anti-divisor, 2.
>
> But I'm not too confident about all this - comments anyone?
> As I said, I wish I understood the motivation for the definition!
>
> Neil
>
>


-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.

-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From maxale at gmail.com  Sat Jul 21 17:37:16 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Sat, 21 Jul 2007 08:37:16 -0700
Subject: definition of anti-divisor
In-Reply-To: <200707211141.l6LBf9dn795643@fry.research.att.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
Message-ID: <d3dac270707210837se098513xafd62ad62bf9360b@mail.gmail.com>

On 7/21/07, N. J. A. Sloane <njas at research.att.com> wrote:

> The term anti-divisor seems to be due to Jon Perry.
> I wish I understood the motivation for the definition!
>
> There are links to various webpages of his, but they are all broken
> and he has not responded to my emails.

Neil, I believe your corrections are correct ;)
Jon Perry's page on anti-divisors is available from archive.org:
http://web.archive.org/web/20070406031719/www.users.globalnet.co.uk/~perry/maths/antidivisor.htm

Max




When I posted my version of the correct definition
of anti-divisor earlier this morning, I should have
given credit to Maximilian Hasler and Max Alekseyev
for saying exactly the same thing in earlier
postings (which I had not read).
Maximilian, Max - my apologies!
Neil




From njas at research.att.com  Sat Jul 21 18:00:32 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Sat, 21 Jul 2007 12:00:32 -0400 (EDT)
Subject: credit for corrected definition of anti-divisor
In-Reply-To: <a851af150707210531s2e01e2b2p3ed2e5c31e635d8d@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <a851af150707210531s2e01e2b2p3ed2e5c31e635d8d@mail.gmail.com>
Message-ID: <200707211600.l6LG0WWW831020@fry.research.att.com>

> According to the second definition, above, an antidivisor must first be a
non-divisor of x. {4, 5, 6, 7} are not non-divisors of 3.

I was assuming that a non-divisor would be <= n by default.

Diana

On 7/21/07, Diana Mecum <diana.mecum at gmail.com> wrote:
>
> Dr. Sloane,
>
> As far as the second definition, and
>
> > %C A130799 Non-divisor: a number k which does not divide a given number
> x. Anti-divisor: a non-divisor k of x with the property that k is an odd
> divisor of 2x-1 or 2x+1, or an even divisor of 2x.
>
> > This definition also fails for n = 3: it gives 5 antidivisors,
> 2,4,5,6,7.
>
> According to the second definition, above, an antidivisor must first be a
> non-divisor of x. {4, 5, 6, 7} are not non-divisors of 3. Therefore, I
> believe that the second definition is correct as it stands.
>
> Diana
>
> On 7/21/07, N. J. A. Sloane <njas at research.att.com> wrote:
> >
> >
> > Dear seqfans, There are currently two versions
> > of the definition of anti-divisor in the OEIS:
> >
> > %C A066272 If an odd number d in the range 1 < d < n divides N
> > where N is any one of 2n-1, 2n or 2n+1
> > then N/d is called an anti-divisor of n.
> >
> > %e A066272 For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd
> > divisors > 1 {5,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the
> > anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 4.
> >
> > But this definition fails for n = 3, as someone mentioned last night.
> > We know from the OEIS that 3 has a single antidivisor, 2.
> > According to this definition 3 has no antidivisors.
> >
> > There is also this program, which I have not checked:
> > %t A066272 antid[ n_ ] := Select[ Union[ Join[ Select[ Divisors[ 2n - 1
> > ], OddQ[ # ] && # != 1 & ], Select[ Divisors[ 2n + 1 ], OddQ[ # ] && # != 1
> > & ], 2n/Select[ Divisors[ 2*n ], OddQ[ # ] && # != 1 & ] ] ] }, # < n & ] ];
> > Table[ Length[ antid[ n ] ], {n, 1, 100} ]
> >
> > The other definition is:
> >
> > %I A130799
> > %S A130799
> > 2,3,2,3,4,2,3,5,3,5,2,6,3,4,7,2,3,7,5,8,2,3,5,9,3,4,9,2,6,10,3,11,
> > %T A130799
> > 2,3,5,7,11,4,5,7,12,2,3,13,3,8,13,2,6,14,3,4,5,9,15,2,3,5,9,15,7,
> > %U A130799
> > 16,2,3,7,10,17,3,4,17,2,5,6,11,18,3,5,8,11,19,2,3,19,4,12,20,2,3,7
> > %N A130799 Triangle read by rows in which row n (n>=3) list the
> > anti-divisors of n.
> > %C A130799 Non-divisor: a number k which does not divide a given number
> > x. Anti-divisor: a non-divisor k of x with the property that k is an odd
> > divisor of 2x-1 or 2x+1, or an even divisor of 2x.
> > %C A130799 There are no anti-divisors of 1 and 2.
> > %e A130799 Anti-divisors of 3 through 20:
> > %e A130799 3: 2
> > %e A130799 4: 3
> > %e A130799 5: 2, 3
> > %e A130799 6: 4
> > %e A130799 7: 2, 3, 5
> > %e A130799 8: 3, 5
> > %e A130799 9: 2, 6
> >
> > This definition also fails for n = 3: it gives 5 antidivisors,
> > 2,4,5,6,7.
> >
> > The term anti-divisor seems to be due to Jon Perry.
> > I wish I understood the motivation for the definition!
> >
> > There are links to various webpages of his, but they are all broken
> > and he has not responded to my emails.
> >
> > It seems to me that both of the above definitions are incorrect, and
> > should be changed to:
> >
> > Definition: If an odd number i in the range 1 < i <= n divides N
> > where N is any one of 2n-1, 2n or 2n+1
> > then d = N/i is called an anti-divisor of n.
> >
> > Equivalently, an anti-divisor of n is a number d in the range [1..n]
> > which does not divide n and is either an odd divisor of 2n-1 or 2n+1,
> > or an even divisor of 2n.
> >
> > Now both definitions seem to work correctly for n=3, giving
> > a single anti-divisor, 2.
> >
> > But I'm not too confident about all this - comments anyone?
> > As I said, I wish I understood the motivation for the definition!
> >
> > Neil
> >
> >
>
>
> --
> "God made the integers, all else is the work of man."
> L. Kronecker, Jahresber. DMV 2, S. 19.




-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From maxale at gmail.com  Sat Jul 21 22:17:53 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Sat, 21 Jul 2007 13:17:53 -0700
Subject: definition of anti-divisor
In-Reply-To: <200707211141.l6LBf9dn795643@fry.research.att.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
Message-ID: <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>

On 7/21/07, N. J. A. Sloane <njas at research.att.com> wrote:

> Definition: If an odd number i in the range 1 < i <= n divides N
> where N is any one of 2n-1, 2n or 2n+1
> then d = N/i is called an anti-divisor of n.
>
> Equivalently, an anti-divisor of n is a number d in the range [1..n]
> which does not divide n and is either an odd divisor of 2n-1 or 2n+1,
> or an even divisor of 2n.

Yet another definition:

k is a non-divisor of n iff:
1 < k < n
and
| (n mod k) - k/2 | <= 1.

Max




From maxale at gmail.com  Sat Jul 21 22:19:27 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Sat, 21 Jul 2007 13:19:27 -0700
Subject: definition of anti-divisor
In-Reply-To: <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
Message-ID: <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>

On 7/21/07, Max Alekseyev <maxale at gmail.com> wrote:

> Yet another definition:
>
> k is a non-divisor of n iff:
> 1 < k < n
> and
> | (n mod k) - k/2 | <= 1.

Oops. It should be

| (n mod k) - k/2 | < 1.

Max




From jvospost3 at gmail.com  Sun Jul 22 08:54:59 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Sat, 21 Jul 2007 23:54:59 -0700
Subject: definition of anti-divisor
In-Reply-To: <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
Message-ID: <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>

Thank you Neil, for putting the cached John Perry Page where we can
get it.  Thank you Max, Diana, et al for corrections and
clarifications.

To extend, then, eliminating the "more" (though anyone is welcome to
submit mathematica or equivalent, and a b-list):

COMMENT FROM Jonathan Vos Post RE A130846

%I A130846
%S A130846 2, 3, 23, 4, 235, 35, 26, 347, 237, 58, 2359, 349, 2610,
311, 235711, 45712, 2313, 3813, 2614, 345915, 235915, 716, 2371017,
3417, 2561118, 3581119, 2319, 41220, 237921, 35791321, 2561322, 3423,
23101423, 824, 2351525, 3457111525, 2671126, 391627, 23927, 45121728,
2351729, 3829, 26710131830, 3471331, 2351931, 51932,  239111433,
349112033
%N A130846 Replace n by the concatenation of its anti-divisors.
%H A130846 Jon Perry, <a
href="http://www.research.att.com/~njas/sequences/a066272a.html">The
Anti-Divisor</a>, cached copy.
%e A130846 a(21)-a(50) adapted from cached Perry page.
%Y A130846 Cf. A037278, A066272, A120712, A106708, A130799.
%O A130846 3
%K A130846 ,base,easy,nonn,
%A A130846 Jonathan Vos Post (jvospost2 at yahoo.com), Jul 22 2007

Next, the matters of iteration, fixed points, primes in A130846 can
commence, subject to Neil's taste as bounded by "less"itude.




From jvospost3 at gmail.com  Sun Jul 22 09:02:55 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Sun, 22 Jul 2007 00:02:55 -0700
Subject: definition of anti-divisor
In-Reply-To: <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
Message-ID: <5542af940707220002l4ca17d60i426f4a982d006d0b@mail.gmail.com>

Even the number of digits of, for n>2, A130846(n) = 1, 1, 2, 1, 3, 2,
2, 3, 3, 2, 4, 3, 4, 3, 6, 5, 4, 4, 4, 6, 6, 3... is nontrivial.

Raiding the spectre of indices n of record values of number of digits
of A130846(n).

I wonder if switching to base 2 or other bases is lessitudinal?  Or,
if tried, are there any interesting things to see?

What is the behavior of A130846(n)/n?

Since I don't really know what Jon Perry was getting at, I don't know
how dumb these questions and derived sequences are, hence I rely on
the judgment of seqfans.




From davidwwilson at comcast.net  Sun Jul 22 09:19:49 2007
From: davidwwilson at comcast.net (David Wilson)
Date: Sun, 22 Jul 2007 03:19:49 -0400
Subject: 919 conjecture
References: <200707171512.AA1076494578@TanyaKhovanova.com>
Message-ID: <004b01c7cc30$b15ae890$6501a8c0@yourxhtr8hvc4p>

The graph of A002375 strongly indicates that the number of Goldbach 
partitions of even n >= 4 remains positive. Similarly, the graph of A014085 
strongly indicates that the number of primes between adjacent positive 
squares remains positive. Though neither conjecture is proved, and a freak 
departure from the visible pattern is theoretically possible, the trends in 
the empirical evidence suggest that both conjectures are a very safe bet. 
Conway suggests calling such conjectures, with ample empirical evidence and 
little reason to doubt that evidence, "sureties". Thus Goldbach's and 
Legendre's conjectures are "sure" if not provably true.

I suggest that you count and graph the number of primes strictly between 
adjacent palindromes. I suspect that, except for anomalous adjacent 
palindrome pairs (10^k-1, 10^k+1) (which cannot be adjacent palindromic 
primes), the number of intervening primes will exhibit a visually convincing 
growth pattern that will convince you that (919, 929) is indeed the last 
pair of consecutive palindromes and consecutive primes.

----- Original Message ----- 
From: "Tanya Khovanova" <tanyakh at TanyaKhovanova.com>
To: <seqfan at ext.jussieu.fr>
Sent: Tuesday, July 17, 2007 6:12 PM
Subject: 919 conjecture


> Hello all,
>
> I was looking at A069803 - Smaller of two consecutive palindromic primes: 
> 2, 3, 5, 7, 181, 787, 919
> Conjectured to be complete.
>
> I am interested in seeing a proof that 919 is actually the largest 
> palindromic prime such that the next prime is palindromic.
> I checked up to 10^8 with Mathematica coding.
>
> Also, it is obvious that the distance from a palindrome n to the next one 
> is more than Sqrt(n/10). It is clear that prime gaps grow slower than 
> that. Looking at the prime gaps sequence A053303, it is easy to prove that 
> 919 is the last number like that up to 10^16.
>
> Is there a bound for prime gaps that proves that the gaps are less than 
> Sqrt(n/10) starting from some n?
>
> Tanya
>
>
> _________________________________________________________________
> Need personalized email and website? Look no further. It's easy
> with Doteasy $0 Web Hosting! Learn more at www.doteasy.com
>
>
> -- 
> No virus found in this incoming message.
> Checked by AVG Free Edition.
> Version: 7.5.476 / Virus Database: 269.10.8/906 - Release Date: 7/17/2007 
> 6:30 PM
> 





From pauldhanna at juno.com  Sun Jul 22 13:21:57 2007
From: pauldhanna at juno.com (Paul D. Hanna)
Date: Sun, 22 Jul 2007 07:21:57 -0400
Subject: A006336 - Unexpected Relation to Golden Ratio?
Message-ID: <20070722.072158.944.1.pauldhanna@juno.com>

Seqfans, 
     Consider the nice sequence A006336: 
a(n) = a(n-1) + a(n-1 - number of even terms so far). 
http://www.research.att.com/~njas/sequences/A006336
begins:
[1,2,3,5,8,11,16,21,29,40,51,67,88,109,138,167,207,258,309,376,...].
 
My COMMENT (NOT submitted to OEIS): 
-----------------------------------------------------------
It seems that A006336 can be generated by a rule using the golden ratio:
 
a(n) = a(n-1) + a([n/Phi]) for n>1 with a(1)=1  where Phi =
(sqrt(5)+1)/2, 
 
i.e., the number of even terms up to position n-1 equals: 
n-1 - [n/Phi] for n>1 where Phi = (sqrt(5)+1)/2. 
 
(PARI): 
a(n) = if(n==1,1, a(n-1) + a( floor(n/((sqrt(5)+1)/2)) )  )
-----------------------------------------------------------
 
Would someone verify if these are indeed equivalent definitions, at least
empirically?  
Or, what is the first position in which terms are NOT equal? 
 
If these are equivalent, then this is another unexpected appearance of
that ubiquitous constant. 
Thanks, 
     Paul 
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From tanyakh at TanyaKhovanova.com  Sun Jul 22 18:16:59 2007
From: tanyakh at TanyaKhovanova.com (Tanya Khovanova)
Date: Sun, 22 Jul 2007 09:16:59 -0700
Subject: sequence joke
In-Reply-To: <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
Message-ID: <200707220916.AA2126905490@TanyaKhovanova.com>

So, extending the table of my 20 July 2007 email:

===========
n  a(n)  factorization
3  2     prime
4  3     prime
5  23    prime
6  4     2^2
7  235   5 * 47
8  35    5 * 7
9  26    2 * 13
10 347   prime
11 237   3 * 79
12 58    2 * 29
13 2359  7 * 337
14 349   prime
15 2610  2 * 3^2 * 5 * 29
16 311   prime
17 235711  7 * 151 * 223
18 45712 2^4 * 2857
19 2313  3^2 * 257
20 3813  3 * 31 * 41
21 2614 2 * 1307
22 345915 3^2 * 5 * 7687
23 235915 5 * 29 * 1627
24 716   2^2 * 179
25 2371017 3 * 11 * 71849
26 3417 3 * 17 * 67
27 2561118 2 * 3 * 7 * 17^2 * 211
28 3581119 37 * 96787
29 2319  3 * 773
30 41220 2^2 * 3^2 * 5 * 229
31 237921 3 * 71 * 1117
32 35791321 37 * 967333
33 2561322 2 * 3 * 17 * 25111
34 3423  3 * 7 * 163
35 23101423 97 * 238159
36 824    2^3 * 103
37 2351525 5^2 * 11 * 17 * 503
38 3457111525 5^2 * 7 * 97 * 203659
39 2671126 2 * 1335563
40 391627 prime
41 23927   71 * 337
42 45121728 2^6 * 3 * 235009
43 2351729  17 * 138337
44 3829  7 * 547
45 26710131830 2 * 5 * 41^2 * 757 * 2099
46 3471331 prime
47 2351931 3 * 523 * 1499
48 51932 2^2 * 12983
49 239111433 3^4 * 11 * 43 * 79^2
50 349112033 12119 * 28807
51 2634  2 * 3 * 439
52 3578152135 5 * 79 * 9058613
53 2357152135 5 * 13 * 36263879
54 41236 2^2 * 13^2 * 61
55 23102237 73 * 316469
56 31637 17 * 1861
57 2562338 2 * 23 * 53 * 1051
58 3459132339 3 * 13 * 1229 * 72169
59 2379131739 3^2 * 877 * 301423
60 7811172440 2^3 * 5 * 195279311
61 231141 3 * 77047
62 3452541 3 * 1150847
...




Dear Seqfans,  My old friend Bernardo Recaman Santos just

%I A116700
%S A116700 12,21,23,31,32,34,41,42,43
%N A116700 "Early bird" numbers: write the natural numbers in a string 12345678910111213....
Sequence gives numbers which occur in the string ahead of their natural place.
%C A116700 Based on an idea by Argentinian puzzle creator Jaime Poniachik, these numbers were
MAA.
%e A116700 "12" appears at the start of the string, ahead of its position after "11", so is a
%K A116700 nonn,base,more,nice,new
%O A116700 1,1
%A A116700 Bernardo Recaman Santos (ignotus(AT)hotmail.com), Jul 22 2007

One needs to be a subscriber to have on-line access to
Math Horizons - can anyone get me a copy of the Martin Gardner
article?

Neil




From njas at research.att.com  Mon Jul 23 05:41:32 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Sun, 22 Jul 2007 23:41:32 -0400 (EDT)
Subject: Early Bird numbers
Message-ID: <200707230341.l6N3fWM9861587@fry.research.att.com>

sent a really nice sequence:
 introduced by Martin Gardner in the November 2005 issue of Math. Horizons, published by the
 member.
Return-Path: <warut822 at gmail.com>
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Message-ID: <482644420707230039r7e3eec65u2c9ec1956bf006e at mail.gmail.com>
Date: Mon, 23 Jul 2007 14:39:19 +0700
From: "Warut Roonguthai" <warut822 at gmail.com>
To: seqfan at ext.jussieu.fr
Subject: Re: Early Bird numbers
In-Reply-To: <200707230341.l6N3fWM9861587 at fry.research.att.com>
MIME-Version: 1.0
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Content-Transfer-Encoding: 7bit
Content-Disposition: inline
References: <200707230341.l6N3fWM9861587 at fry.research.att.com>
X-Greylist: IP, sender and recipient auto-whitelisted, not delayed by milter-greylist-3.0 (shiva.jussieu.fr [134.157.0.166]); Mon, 23 Jul 2007 09:39:20 +0200 (CEST)
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X-Virus-Status: Clean
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X-Miltered: at shiva.jussieu.fr with ID 46A45B28.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr)!

Here are the first 157 terms of the early bird sequence:

12,21,23,31,32,34,41,42,43,45,51,52,53,54,56,61,62,63,64,65,
67,71,72,73,74,75,76,78,81,82,83,84,85,86,87,89,91,92,93,94,
95,96,97,98,99,101,110,111,112,121,122,123,131,132,141,142,
151,152,161,162,171,172,181,182,191,192,201,202,210,211,
212,213,214,215,216,217,218,219,220,221,222,223,231,232,
233,234,241,242,243,251,252,253,261,262,263,271,272,273,
281,282,283,291,292,293,301,302,303,310,311,312,313,314,
315,316,317,318,319,320,321,322,323,324,325,326,327,328,
329,330,331,332,333,334,341,342,343,344,345,351,352,353,
354,361,362,363,364,371,372,373,374,381,382,383,384,391,
392,393,394

Note that the natural place of n begins at

dn + 1 - (10^d - 1)/9

where d is the number of decimal digits of n, i.e.,

d = floor(log10(n)) + 1

Can this be a new sequence too?

BTW, I don't have access to Martin Gardner's article.

Warut

On 7/23/07, N. J. A. Sloane <njas at research.att.com> wrote:
>
> Dear Seqfans,  My old friend Bernardo Recaman Santos just
> sent a really nice sequence:
>
> %I A116700
> %S A116700 12,21,23,31,32,34,41,42,43
> %N A116700 "Early bird" numbers: write the natural numbers in a string 12345678910111213....
> Sequence gives numbers which occur in the string ahead of their natural place.
> %C A116700 Based on an idea by Argentinian puzzle creator Jaime Poniachik, these numbers were
>  introduced by Martin Gardner in the November 2005 issue of Math. Horizons, published by the
> MAA.
> %e A116700 "12" appears at the start of the string, ahead of its position after "11", so is a
>  member.
> %K A116700 nonn,base,more,nice,new
> %O A116700 1,1
> %A A116700 Bernardo Recaman Santos (ignotus(AT)hotmail.com), Jul 22 2007
>
> One needs to be a subscriber to have on-line access to
> Math Horizons - can anyone get me a copy of the Martin Gardner
> article?
>
> Neil
>




From jeremy.gardiner at btinternet.com  Mon Jul 23 09:45:37 2007
From: jeremy.gardiner at btinternet.com (JEREMY GARDINER)
Date: Mon, 23 Jul 2007 08:45:37 +0100 (BST)
Subject: Early Bird numbers
In-Reply-To: <200707230341.l6N3fWM9861587@fry.research.att.com>
Message-ID: <886534.93581.qm@web86607.mail.ukl.yahoo.com>

 
  I found some interesting references through Google:
   
  http://www.itsoc.org/publications/nltr/it1202.pdf
http://www.itsoc.org/publications/nltr/it0303web.pdf
http://membership.kcatm.org/pub/summ2-06.pdf
   
  Jeremy
   
  ----------------------------------------------------------------------------------------

"N. J. A. Sloane" <njas at research.att.com> wrote:
  
Dear Seqfans, My old friend Bernardo Recaman Santos just
sent a really nice sequence:

%I A116700
%S A116700 12,21,23,31,32,34,41,42,43
%N A116700 "Early bird" numbers: write the natural numbers in a string 12345678910111213....
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From warut822 at gmail.com  Mon Jul 23 14:35:22 2007
From: warut822 at gmail.com (Warut Roonguthai)
Date: Mon, 23 Jul 2007 19:35:22 +0700
Subject: Early Bird numbers
In-Reply-To: <450839.70489.qm@web86611.mail.ukl.yahoo.com>
References: <482644420707230039r7e3eec65u2c9ec1956bf006e@mail.gmail.com>
	 <450839.70489.qm@web86611.mail.ukl.yahoo.com>
Message-ID: <482644420707230535h41b06b78u11273f7c8804bf41@mail.gmail.com>

FYI, here's my Ubasic program for generating the early bird sequence:

10   X=""
20   for N=1 to 396
30   A=cutspc(str(N))
40   if instr(X,A)>0 then print N;
50   X+=A
60   next N

Warut




From jeremy.gardiner at btinternet.com  Mon Jul 23 13:53:35 2007
From: jeremy.gardiner at btinternet.com (JEREMY GARDINER)
Date: Mon, 23 Jul 2007 12:53:35 +0100 (BST)
Subject: Early Bird numbers
In-Reply-To: <482644420707230039r7e3eec65u2c9ec1956bf006e@mail.gmail.com>
Message-ID: <450839.70489.qm@web86611.mail.ukl.yahoo.com>

I checked the parity of Warut's early bird values and there *may* be a connection with the following sequences: I have no idea why this should be the case and my apology if this is a red herring !!!
   
  A091264  Matrix defined by a(n,k) = 2^n + (k-1), read by antidiagonals.
A128138  A000012 * A128132.
A128219  A000012 * A127701. a(1) = 1, a(2) = 2, a(3) = 2; by rows, n-1 terms of 2, 3, 4...followed by "n".
   
  Parity values below; note these fail to match in a couple of places, suggesting I may have made an editing error, or that perhaps Warut's values should be checked?
   
  Jeremy
           0,1,1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,
A091264 ,0,1,1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0, 
           1,0,0,0,1,1,0,1,0,0,0,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,
A128138 ,1,0,0,0,1,1,0,1,0,0,0,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0, 
A128219 ,1,0,0,0,1,1,0,1,0,0,0,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0, 
  A091264  Matrix defined by a(n,k) = 2^n + (k-1), read by antidiagonals.
A128138  A000012 * A128132.
A128219  A000012 * A127701. a(1) = 1, a(2) = 2, a(3) = 2; by rows, n-1 terms of 2, 3, 4...followed by "n".
   
  ------------------------------------------------------------------------------------------
Warut Roonguthai <warut822 at gmail.com> wrote:
  Here are the first 157 terms of the early bird sequence:

12,21,23,31,32,34,41,42,43,45,51,52,53,54,56,61,62,63,64,65,
67,71,72,73,74,75,76,78,81,82,83,84,85,86,87,89,91,92,93,94,
95,96,97,98,99,101,110,111,112,121,122,123,131,132,141,142,
151,152,161,162,171,172,181,182,191,192,201,202,210,211,
212,213,214,215,216,217,218,219,220,221,222,223,231,232,
233,234,241,242,243,251,252,253,261,262,263,271,272,273,
281,282,283,291,292,293,301,302,303,310,311,312,313,314,
315,316,317,318,319,320,321,322,323,324,325,326,327,328,
329,330,331,332,333,334,341,342,343,344,345,351,352,353,
354,361,362,363,364,371,372,373,374,381,382,383,384,391,
392,393,394

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From joshua.zucker at gmail.com  Mon Jul 23 15:58:50 2007
From: joshua.zucker at gmail.com (Joshua Zucker)
Date: Mon, 23 Jul 2007 06:58:50 -0700
Subject: Early Bird numbers
In-Reply-To: <482644420707230535h41b06b78u11273f7c8804bf41@mail.gmail.com>
References: <482644420707230039r7e3eec65u2c9ec1956bf006e@mail.gmail.com>
	 <450839.70489.qm@web86611.mail.ukl.yahoo.com>
	 <482644420707230535h41b06b78u11273f7c8804bf41@mail.gmail.com>
Message-ID: <721e81490707230658g33f6d925k8d0c7d0f7b80bf4e@mail.gmail.com>

I wrote my own program and let it run to make all the terms up to
1000.  Up to 394 they match the terms Warut's program produced.

--Joshua Zucker

12 21 23 31 32 34 41 42 43 45 51 52 53 54 56 61 62 63 64 65 67 71 72
73 74 75 76 78 81 82 83 84 85 86 87 89 91 92 93 94 95 96 97 98 99 101
110 111 112 121 122 123 131 132 141 142 151 152 161 162 171 172 181
182 191 192 201 202 210 211 212 213 214 215 216 217 218 219 220 221
222 223 231 232 233 234 241 242 243 251 252 253 261 262 263 271 272
273 281 282 283 291 292 293 301 302 303 310 311 312 313 314 315 316
317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333
334 341 342 343 344 345 351 352 353 354 361 362 363 364 371 372 373
374 381 382 383 384 391 392 393 394 401 402 403 404 410 411 412 413
414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430
431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 451 452
453 454 455 456 461 462 463 464 465 471 472 473 474 475 481 482 483
484 485 491 492 493 494 495 501 502 503 504 505 510 511 512 513 514
515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531
532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548
549 550 551 552 553 554 555 556 561 562 563 564 565 566 567 571 572
573 574 575 576 581 582 583 584 585 586 591 592 593 594 595 596 601
602 603 604 605 606 610 611 612 613 614 615 616 617 618 619 620 621
622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638
639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655
656 657 658 659 660 661 662 663 664 665 666 667 671 672 673 674 675
676 677 678 681 682 683 684 685 686 687 691 692 693 694 695 696 697
701 702 703 704 705 706 707 710 711 712 713 714 715 716 717 718 719
720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736
737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753
754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770
771 772 773 774 775 776 777 778 781 782 783 784 785 786 787 788 789
791 792 793 794 795 796 797 798 801 802 803 804 805 806 807 808 810
811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827
828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844
845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861
862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878
879 880 881 882 883 884 885 886 887 888 889 891 892 893 894 895 896
897 898 899 901 902 903 904 905 906 907 908 909 910 911 912 913 914
915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931
932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948
949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965
966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982
983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999

On 7/23/07, Warut Roonguthai <warut822 at gmail.com> wrote:
> FYI, here's my Ubasic program for generating the early bird sequence:
>
> 10   X=""
> 20   for N=1 to 396
> 30   A=cutspc(str(N))
> 40   if instr(X,A)>0 then print N;
> 50   X+=A
> 60   next N
>
> Warut
>




From maximilian.hasler at gmail.com  Mon Jul 23 18:00:24 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Mon, 23 Jul 2007 12:00:24 -0400
Subject: Early Bird numbers
In-Reply-To: <721e81490707230658g33f6d925k8d0c7d0f7b80bf4e@mail.gmail.com>
References: <482644420707230039r7e3eec65u2c9ec1956bf006e@mail.gmail.com>
	 <450839.70489.qm@web86611.mail.ukl.yahoo.com>
	 <482644420707230535h41b06b78u11273f7c8804bf41@mail.gmail.com>
	 <721e81490707230658g33f6d925k8d0c7d0f7b80bf4e@mail.gmail.com>
Message-ID: <3c3af2330707230900y56c82727tfae0564679ae8746@mail.gmail.com>

This is another example of how computers can denature things.
"Early birds" seem interesting when you do it by hand, for n<199, say
; but from then on it would be more interesting to study the
complementary sequence ("late birds" ? which appear not before their
"regular" place) - e.g. among 500..999 only about 50 terms are not
present.
At a first glance, I'd almost be tempted to conjecture that from a
certain rank on, the only late birds are of the form  d*10^k with d in
{1,...,9 }, and all other numbers are early birds...
M.H.

On 7/23/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
> I wrote my own program and let it run to make all the terms up to
> 1000.  Up to 394 they match the terms Warut's program produced.
>
> --Joshua Zucker
>
> 12 21 23 31 32 34 41 42 43 45 51 52 53 54 56 61 62 63 64 65 67 71 72
> 73 74 75 76 78 81 82 83 84 85 86 87 89 91 92 93 94 95 96 97 98 99 101
> 110 111 112 121 122 123 131 132 141 142 151 152 161 162 171 172 181
> 182 191 192 201 202 210 211 212 213 214 215 216 217 218 219 220 221
> 222 223 231 232 233 234 241 242 243 251 252 253 261 262 263 271 272
> 273 281 282 283 291 292 293 301 302 303 310 311 312 313 314 315 316
> 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333
> 334 341 342 343 344 345 351 352 353 354 361 362 363 364 371 372 373
> 374 381 382 383 384 391 392 393 394 401 402 403 404 410 411 412 413
> 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430
> 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 451 452
> 453 454 455 456 461 462 463 464 465 471 472 473 474 475 481 482 483
> 484 485 491 492 493 494 495 501 502 503 504 505 510 511 512 513 514
> 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531
> 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548
> 549 550 551 552 553 554 555 556 561 562 563 564 565 566 567 571 572
> 573 574 575 576 581 582 583 584 585 586 591 592 593 594 595 596 601
> 602 603 604 605 606 610 611 612 613 614 615 616 617 618 619 620 621
> 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638
> 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655
> 656 657 658 659 660 661 662 663 664 665 666 667 671 672 673 674 675
> 676 677 678 681 682 683 684 685 686 687 691 692 693 694 695 696 697
> 701 702 703 704 705 706 707 710 711 712 713 714 715 716 717 718 719
> 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736
> 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753
> 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770
> 771 772 773 774 775 776 777 778 781 782 783 784 785 786 787 788 789
> 791 792 793 794 795 796 797 798 801 802 803 804 805 806 807 808 810
> 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827
> 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844
> 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861
> 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878
> 879 880 881 882 883 884 885 886 887 888 889 891 892 893 894 895 896
> 897 898 899 901 902 903 904 905 906 907 908 909 910 911 912 913 914
> 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931
> 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948
> 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965
> 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982
> 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999
>
> On 7/23/07, Warut Roonguthai <warut822 at gmail.com> wrote:
> > FYI, here's my Ubasic program for generating the early bird sequence:
> >
> > 10   X=""
> > 20   for N=1 to 396
> > 30   A=cutspc(str(N))
> > 40   if instr(X,A)>0 then print N;
> > 50   X+=A
> > 60   next N
> >
> > Warut
> >
>




From warut822 at gmail.com  Mon Jul 23 18:10:00 2007
From: warut822 at gmail.com (Warut Roonguthai)
Date: Mon, 23 Jul 2007 23:10:00 +0700
Subject: A006336 - Unexpected Relation to Golden Ratio?
In-Reply-To: <20070722.072158.944.1.pauldhanna@juno.com>
References: <20070722.072158.944.1.pauldhanna@juno.com>
Message-ID: <482644420707230910s22b233aeg6b06cf2dcf0d2fe7@mail.gmail.com>

Very interesting observation, Paul. Your conjecture is too beautiful
to be wrong! However, it seems to be very difficult to prove. I've
checked the first 65,000 terms, but I think it would be easy for some
seqfan programmers to extend the calculation to 10,000,000 terms or so
with C or some other powerful tools. Note that we only have to keep
track of the parity of a(n), not its entire value.

Warut

On 7/22/07, Paul D. Hanna <pauldhanna at juno.com> wrote:
>
> Seqfans,
>      Consider the nice sequence A006336:
> a(n) = a(n-1) + a(n-1 - number of even terms so far).
> http://www.research.att.com/~njas/sequences/A006336
> begins:
> [1,2,3,5,8,11,16,21,29,40,51,67,88,109,138,167,207,258,309,376,...].
>
> My COMMENT (NOT submitted to OEIS):
> -----------------------------------------------------------
> It seems that A006336 can be generated by a rule using the golden ratio:
>
> a(n) = a(n-1) + a([n/Phi]) for n>1 with a(1)=1  where Phi = (sqrt(5)+1)/2,
>
> i.e., the number of even terms up to position n-1 equals:
> n-1 - [n/Phi] for n>1 where Phi = (sqrt(5)+1)/2.
>
> (PARI):
> a(n) = if(n==1,1, a(n-1) + a( floor(n/((sqrt(5)+1)/2)) )  )
> -----------------------------------------------------------
>
> Would someone verify if these are indeed equivalent definitions, at least
> empirically?
> Or, what is the first position in which terms are NOT equal?
>
> If these are equivalent, then this is another unexpected appearance of that
> ubiquitous constant.
> Thanks,
>      Paul




From Eric.Angelini at kntv.be  Mon Jul 23 18:25:48 2007
From: Eric.Angelini at kntv.be (Eric Angelini)
Date: Mon, 23 Jul 2007 18:25:48 +0200
Subject: Divisor d is the total number of divisors
Message-ID: <F05775148D000C4B9321A5113D53CB7D405F91@KNTV-SERVER01.kntv.local>



Hello SeqFans,

Is this seq of interest? 
If yes could someone check and compute a few more terms?

1,8,9,12,18,24,36,...

Integers I having one divisor which is also the total number of divisors of I.

 1 has 1 divisor which is 1
 8 has 4 divs and 4 is one of them
 9 has 3 divs and 3 is one of them
12 has 6 divs and 6 is one of them
18 has 6 divs and 6 is one of them
24 has 8 divs and 8 is one of them
36 has 9 divs and 9 is one of them
...

30 is not a member because 30 has 8 divs but not 8 itself : [1,2,3,5,6,10,15,30]

Best,
?.








From Eric.Angelini at kntv.be  Mon Jul 23 18:33:29 2007
From: Eric.Angelini at kntv.be (Eric Angelini)
Date: Mon, 23 Jul 2007 18:33:29 +0200
Subject: Divisor d is the total number of divisors
Message-ID: <F05775148D000C4B9321A5113D53CB7D405F92@KNTV-SERVER01.kntv.local>

 

Sorry, this is A033950
Best,
?.

(had forgotten "2" in my list -- no hit then in the OEIS :-(



-----Message d'origine-----
De : Eric Angelini 
Envoy? : lundi 23 juillet 2007 18:26
? : seqfan at ext.jussieu.fr
Objet : Divisor d is the total number of divisors



Hello SeqFans,

Is this seq of interest? 
If yes could someone check and compute a few more terms?

1,8,9,12,18,24,36,...

Integers I having one divisor which is also the total number of divisors of I.

 1 has 1 divisor which is 1
 8 has 4 divs and 4 is one of them
 9 has 3 divs and 3 is one of them
12 has 6 divs and 6 is one of them
18 has 6 divs and 6 is one of them
24 has 8 divs and 8 is one of them
36 has 9 divs and 9 is one of them
...

30 is not a member because 30 has 8 divs but not 8 itself : [1,2,3,5,6,10,15,30]

Best,
?.








From maximilian.hasler at gmail.com  Mon Jul 23 18:28:27 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Mon, 23 Jul 2007 12:28:27 -0400
Subject: Early Bird numbers
In-Reply-To: <3c3af2330707230900y56c82727tfae0564679ae8746@mail.gmail.com>
References: <482644420707230039r7e3eec65u2c9ec1956bf006e@mail.gmail.com>
	 <450839.70489.qm@web86611.mail.ukl.yahoo.com>
	 <482644420707230535h41b06b78u11273f7c8804bf41@mail.gmail.com>
	 <721e81490707230658g33f6d925k8d0c7d0f7b80bf4e@mail.gmail.com>
	 <3c3af2330707230900y56c82727tfae0564679ae8746@mail.gmail.com>
Message-ID: <3c3af2330707230928g2bbb5ce9u165e91d66b0e3eef@mail.gmail.com>

at the second glance, I take back my premature proposal concerning a
"late bird" conjecture - as shows the following PHP 1-liner,

Late birds:
<?php $s="."; for(; ++$i < 2000; $s .= $i) if( !strpos($s,"$i")) echo $i,", ";

there are many late birds in the "1xxxx" range for any number of digits.
Sorry...
M.H.

Late birds:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 22,
24, 25, 26, 27, 28, 29, 30, 33, 35, 36, 37, 38, 39, 40, 44, 46, 47,
48, 49, 50, 55, 57, 58, 59, 60, 66, 68, 69, 70, 77, 79, 80, 88, 90,
100, 102, 103, 104, 105, 106, 107, 108, 109, 113, 114, 115, 116, 117,
118, 119, 120, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136,
137, 138, 139, 140, 143, 144, 145, 146, 147, 148, 149, 150, 153, 154,
155, 156, 157, 158, 159, 160, 163, 164, 165, 166, 167, 168, 169, 170,
173, 174, 175, 176, 177, 178, 179, 180, 183, 184, 185, 186, 187, 188,
189, 190, 193, 194, 195, 196, 197, 198, 199, 200, 203, 204, 205, 206,
207, 208, 209, 224, 225, 226, 227, 228, 229, 230, 235, 236, 237, 238,
239, 240, 244, 245, 246, 247, 248, 249, 250, 254, 255, 256, 257, 258,
259, 260, 264, 265, 266, 267, 268, 269, 270, 274, 275, 276, 277, 278,
279, 280, 284, 285, 286, 287, 288, 289, 290, 294, 295, 296, 297, 298,
299, 300, 304, 305, 306, 307, 308, 309, 335, 336, 337, 338, 339, 340,
346, 347, 348, 349, 350, 355, 356, 357, 358, 359, 360, 365, 366, 367,
368, 369, 370, 375, 376, 377, 378, 379, 380, 385, 386, 387, 388, 389,
390, 395, 396, 397, 398, 399, 400, 405, 406, 407, 408, 409, 446, 447,
448, 449, 450, 457, 458, 459, 460, 466, 467, 468, 469, 470, 476, 477,
478, 479, 480, 486, 487, 488, 489, 490, 496, 497, 498, 499, 500, 506,
507, 508, 509, 557, 558, 559, 560, 568, 569, 570, 577, 578, 579, 580,
587, 588, 589, 590, 597, 598, 599, 600, 607, 608, 609, 668, 669, 670,
679, 680, 688, 689, 690, 698, 699, 700, 708, 709, 779, 780, 790, 799,
800, 809, 890, 900, 1000, 1002, 1003, 1004, 1005, 1006, 1007, 1008,
1009, 1010, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1019, 1020,
1022, 1023, ...

On 7/23/07, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:
> This is another example of how computers can denature things.
> "Early birds" seem interesting when you do it by hand, for n<199, say
> ; but from then on it would be more interesting to study the
> complementary sequence ("late birds" ? which appear not before their
> "regular" place) - e.g. among 500..999 only about 50 terms are not
> present.
> At a first glance, I'd almost be tempted to conjecture that from a
> certain rank on, the only late birds are of the form  d*10^k with d in
> {1,...,9 }, and all other numbers are early birds...
> M.H.
>
> On 7/23/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
> > I wrote my own program and let it run to make all the terms up to
> > 1000.  Up to 394 they match the terms Warut's program produced.
> >
> > --Joshua Zucker
> >
> > 12 21 23 31 32 34 41 42 43 45 51 52 53 54 56 61 62 63 64 65 67 71 72
> > 73 74 75 76 78 81 82 83 84 85 86 87 89 91 92 93 94 95 96 97 98 99 101
> > 110 111 112 121 122 123 131 132 141 142 151 152 161 162 171 172 181
> > 182 191 192 201 202 210 211 212 213 214 215 216 217 218 219 220 221
> > 222 223 231 232 233 234 241 242 243 251 252 253 261 262 263 271 272
> > 273 281 282 283 291 292 293 301 302 303 310 311 312 313 314 315 316
> > 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333
> > 334 341 342 343 344 345 351 352 353 354 361 362 363 364 371 372 373
> > 374 381 382 383 384 391 392 393 394 401 402 403 404 410 411 412 413
> > 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430
> > 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 451 452
> > 453 454 455 456 461 462 463 464 465 471 472 473 474 475 481 482 483
> > 484 485 491 492 493 494 495 501 502 503 504 505 510 511 512 513 514
> > 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531
> > 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548
> > 549 550 551 552 553 554 555 556 561 562 563 564 565 566 567 571 572
> > 573 574 575 576 581 582 583 584 585 586 591 592 593 594 595 596 601
> > 602 603 604 605 606 610 611 612 613 614 615 616 617 618 619 620 621
> > 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638
> > 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655
> > 656 657 658 659 660 661 662 663 664 665 666 667 671 672 673 674 675
> > 676 677 678 681 682 683 684 685 686 687 691 692 693 694 695 696 697
> > 701 702 703 704 705 706 707 710 711 712 713 714 715 716 717 718 719
> > 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736
> > 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753
> > 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770
> > 771 772 773 774 775 776 777 778 781 782 783 784 785 786 787 788 789
> > 791 792 793 794 795 796 797 798 801 802 803 804 805 806 807 808 810
> > 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827
> > 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844
> > 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861
> > 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878
> > 879 880 881 882 883 884 885 886 887 888 889 891 892 893 894 895 896
> > 897 898 899 901 902 903 904 905 906 907 908 909 910 911 912 913 914
> > 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931
> > 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948
> > 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965
> > 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982
> > 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999
> >
> > On 7/23/07, Warut Roonguthai <warut822 at gmail.com> wrote:
> > > FYI, here's my Ubasic program for generating the early bird sequence:
> > >
> > > 10   X=""
> > > 20   for N=1 to 396
> > > 30   A=cutspc(str(N))
> > > 40   if instr(X,A)>0 then print N;
> > > 50   X+=A
> > > 60   next N
> > >
> > > Warut
> > >
> >
>




From rayjchandler at sbcglobal.net  Mon Jul 23 18:31:23 2007
From: rayjchandler at sbcglobal.net (Ray Chandler)
Date: Mon, 23 Jul 2007 11:31:23 -0500
Subject: Divisor d is the total number of divisors
In-Reply-To: <F05775148D000C4B9321A5113D53CB7D405F91@KNTV-SERVER01.kntv.local>
References: <F05775148D000C4B9321A5113D53CB7D405F91@KNTV-SERVER01.kntv.local>
Message-ID: <021101c7cd46$e9b55b50$6600000a@HPm400y>

> 
> Is this seq of interest? 
> If yes could someone check and compute a few more terms?
> 
> 1,8,9,12,18,24,36,...
> > 
> Best,
> ?.
> 
Apparently a number of people thought so, see A033950.  (2 is also in the sequence).
Ray





>Is this seq of interest?
>If yes could someone check and compute a few more terms?
>
>1,8,9,12,18,24,36,...
>
>Integers I having one divisor which is also the total number of divisors of I.
>
> 1 has 1 divisor which is 1
> 8 has 4 divs and 4 is one of them
> 9 has 3 divs and 3 is one of them
>12 has 6 divs and 6 is one of them
>18 has 6 divs and 6 is one of them
>24 has 8 divs and 8 is one of them
>36 has 9 divs and 9 is one of them
>...
>
>30 is not a member because 30 has 8 divs but not 8 itself :
>[1,2,3,5,6,10,15,30]

See A033950.

Tony




From noe at sspectra.com  Mon Jul 23 18:30:49 2007
From: noe at sspectra.com (T. D. Noe)
Date: Mon, 23 Jul 2007 09:30:49 -0700
Subject: Divisor d is the total number of divisors
In-Reply-To:  <F05775148D000C4B9321A5113D53CB7D405F91@KNTV-SERVER01.kntv.local>
References:  <F05775148D000C4B9321A5113D53CB7D405F91@KNTV-SERVER01.kntv.local>
Message-ID: <p06240800c2ca880aa512@[192.168.1.43]>

------=_Part_151547_12232578.1185210638925
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Content-Transfer-Encoding: 7bit
Content-Disposition: inline

Following Neil's suggestion to add the complement and his suggestion
for naming it "punctual birds" :

Subject: PRE-NUMBERED NEW SEQUENCE A131881 FROM Maximilian F. Hasler

%I A131881
%S A131881 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18,
19, 20, 22, 24, 25, 26, 27,
28, 29, 30, 33, 35, 36, 37, 38, 39, 40, 44, 46, 47, 48, 49, 50, 55,
57, 58, 59, 60,
66, 68, 69, 70, 77, 79, 80, 88, 90, 100, 102, 103, 104, 105, 106, 107,
108, 109, 113
%N A131881 "punctual birds" - numbers which are not in A116700
%C A131881 Numbers n that do not occur in the concatenation of 1,2,3...,n-1.
There is no punctual bird larger than 9*10^k and smaller than
10^(k+1), for any integer k.
%D A131881 Gardner, Martin. (November 2005). Transcendentals and early birds.
Math Horizons XIII(2), pp 5, 34.
%H A131881 <a href="http://www.itsoc.org/publications/nltr/it1202.pdf">Solomon
W. Golomb, "EARLY BIRD NUMBERS", in IEEE Information TheorySociety
Newsletter, Vol. 52, No. 4, December 2002</a>
%H A131881 <a href="http://membership.kcatm.org/pub/summ2-06.pdf">R.
Barger (editor), "Brain Teaser" in Volume 1, Issue 1 of KANSAS CITY
AREA TEACHERS OF MATHEMATICS"</a>
%e A131881 The first number not in this sequence is the early bird
"12" which occurs as concatenation of 1 and 2.
%o A131881 < ?php $s="."; for(; ++$i < 2000; $s .= $i )
if(!strpos($s,"$i")) echo $i,", ";
%Y A131881 Cf. A116700 (early birds)
%O A131881 1
%K A131881 ,base,easy,nonn,
%A A131881 Maximilian F. Hasler (Maximilian.Hasler at gmail.com), Jul 23 2007

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MTEKMTIgMTMKMTMgMTQKMTQgMTUKMTUgMTYKMTYgMTcKMTcgMTgKMTggMTkK
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NTUKNDIgNTcKNDMgNTgKNDQgNTkKNDUgNjAKNDYgNjYKNDcgNjgKNDggNjkK
NDkgNzAKNTAgNzcKNTEgNzkKNTIgODAKNTMgODgKNTQgOTAKNTUgMTAwCjU2
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------=_Part_151547_12232578.1185210638925--




From keynews.tv at skynet.be  Mon Jul 23 19:17:28 2007
From: keynews.tv at skynet.be (Eric Angelini)
Date: Mon, 23 Jul 2007 19:17:28 +0200
Subject: Early Bird numbers
In-Reply-To: <721e81490707230658g33f6d925k8d0c7d0f7b80bf4e@mail.gmail.com>
Message-ID: <!&!AAAAAAAAAAAuAAAAAAAAACCO5uoaeU1MslQniZBHPHIBAPBXdRSNAAxLkyGlET1Ty30AAAADg9wAABAAAACvZHg3kvjSTaKgcanQ9RA1AQAAAAA=@skynet.be>


What about doing recursively the same with the "early bird" seq? 

We would then have to compare the concatenation of the "early bird" 
integers to 12345678910111213141516... and ask ourselves "what integers
come ahead now?"

By hand I get this (N = naturals, EB = Early Birds):

 
N=12345678910111213141516171819202122232425262728293031323334353637383940...
EB=122123313234414243455152535456616263646567717273747576788182838485868789.
..

EB(2)=12,13,14,21,22,23,24,25,26,31,32,33,34,35,36,37,38,41,42,43,44,45,
      46,47,48,51...

We could start again from there to compute EB(3), etc.

Could we bump into a "fixed point seq"? What would that mean?

Best,
?.



-----Message d'origine-----
De : Joshua Zucker [mailto:joshua.zucker at gmail.com] 
Envoy? : lundi 23 juillet 2007 15:59
? : Warut Roonguthai
Cc : seqfan at ext.jussieu.fr
Objet : Re: Early Bird numbers

I wrote my own program and let it run to make all the terms up to
1000.  Up to 394 they match the terms Warut's program produced.

--Joshua Zucker

12 21 23 31 32 34 41 42 43 45 51 52 53 54 56 61 62 63 64 65 67 71 72
73 74 75 76 78 81 82 83 84 85 86 87 89 91 92 93 94 95 96 97 98 99 101
110 111 112 121 122 123 131 132 141 142 151 152 161 162 171 172 181
182 191 192 201 202 210 211 212 213 214 215 216 217 218 219 220 221
222 223 231 232 233 234 241 242 243 251 252 253 261 262 263 271 272
273 281 282 283 291 292 293 301 302 303 310 311 312 313 314 315 316
317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333
334 341 342 343 344 345 351 352 353 354 361 362 363 364 371 372 373
374 381 382 383 384 391 392 393 394 401 402 403 404 410 411 412 413
414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430
431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 451 452
453 454 455 456 461 462 463 464 465 471 472 473 474 475 481 482 483
484 485 491 492 493 494 495 501 502 503 504 505 510 511 512 513 514
515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531
532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548
549 550 551 552 553 554 555 556 561 562 563 564 565 566 567 571 572
573 574 575 576 581 582 583 584 585 586 591 592 593 594 595 596 601
602 603 604 605 606 610 611 612 613 614 615 616 617 618 619 620 621
622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638
639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655
656 657 658 659 660 661 662 663 664 665 666 667 671 672 673 674 675
676 677 678 681 682 683 684 685 686 687 691 692 693 694 695 696 697
701 702 703 704 705 706 707 710 711 712 713 714 715 716 717 718 719
720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736
737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753
754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770
771 772 773 774 775 776 777 778 781 782 783 784 785 786 787 788 789
791 792 793 794 795 796 797 798 801 802 803 804 805 806 807 808 810
811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827
828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844
845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861
862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878
879 880 881 882 883 884 885 886 887 888 889 891 892 893 894 895 896
897 898 899 901 902 903 904 905 906 907 908 909 910 911 912 913 914
915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931
932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948
949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965
966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982
983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999

On 7/23/07, Warut Roonguthai <warut822 at gmail.com> wrote:
> FYI, here's my Ubasic program for generating the early bird sequence:
>
> 10   X=""
> 20   for N=1 to 396
> 30   A=cutspc(str(N))
> 40   if instr(X,A)>0 then print N;
> 50   X+=A
> 60   next N
>
> Warut
>






From maximilian.hasler at gmail.com  Mon Jul 23 19:17:23 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Mon, 23 Jul 2007 13:17:23 -0400
Subject: Early Bird numbers
In-Reply-To: <3c3af2330707231010g931f3cfl50118a3a07020f3b@mail.gmail.com>
References: <482644420707230039r7e3eec65u2c9ec1956bf006e@mail.gmail.com>
	 <450839.70489.qm@web86611.mail.ukl.yahoo.com>
	 <482644420707230535h41b06b78u11273f7c8804bf41@mail.gmail.com>
	 <721e81490707230658g33f6d925k8d0c7d0f7b80bf4e@mail.gmail.com>
	 <200707231623.l6NGNZMN1224353@fry.research.att.com>
	 <200707231659.l6NGxGe81231836@fry.research.att.com>
	 <3c3af2330707231010g931f3cfl50118a3a07020f3b@mail.gmail.com>
Message-ID: <3c3af2330707231017j1b99c155ga7514f3430c3b302@mail.gmail.com>

Definitely, I did too much administrative work this morning.
As the last line of my B-file shows, the comment is wrong.
Once again public apologies and Neil, please delete the 2nd line of the %C...
(9090 is a punctual bird.)
Maximilian

On 7/23/07, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:
> Following Neil's suggestion to add the complement and his suggestion
> for naming it "punctual birds" :
>
> Subject: PRE-NUMBERED NEW SEQUENCE A131881 FROM Maximilian F. Hasler
>
> %I A131881
> %S A131881 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18,
> 19, 20, 22, 24, 25, 26, 27,
> 28, 29, 30, 33, 35, 36, 37, 38, 39, 40, 44, 46, 47, 48, 49, 50, 55,
> 57, 58, 59, 60,
> 66, 68, 69, 70, 77, 79, 80, 88, 90, 100, 102, 103, 104, 105, 106, 107,
> 108, 109, 113
> %N A131881 "punctual birds" - numbers which are not in A116700
> %C A131881 Numbers n that do not occur in the concatenation of 1,2,3...,n-1.
> There is no punctual bird larger than 9*10^k and smaller than
> 10^(k+1), for any integer k.
> %D A131881 Gardner, Martin. (November 2005). Transcendentals and early birds.
> Math Horizons XIII(2), pp 5, 34.
> %H A131881 <a href="http://www.itsoc.org/publications/nltr/it1202.pdf">Solomon
> W. Golomb, "EARLY BIRD NUMBERS", in IEEE Information TheorySociety
> Newsletter, Vol. 52, No. 4, December 2002</a>
> %H A131881 <a href="http://membership.kcatm.org/pub/summ2-06.pdf">R.
> Barger (editor), "Brain Teaser" in Volume 1, Issue 1 of KANSAS CITY
> AREA TEACHERS OF MATHEMATICS"</a>
> %e A131881 The first number not in this sequence is the early bird
> "12" which occurs as concatenation of 1 and 2.
> %o A131881 < ?php $s="."; for(; ++$i < 2000; $s .= $i )
> if(!strpos($s,"$i")) echo $i,", ";
> %Y A131881 Cf. A116700 (early birds)
> %O A131881 1
> %K A131881 ,base,easy,nonn,
> %A A131881 Maximilian F. Hasler (Maximilian.Hasler at gmail.com), Jul 23 2007




From maxale at gmail.com  Mon Jul 23 19:29:13 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Mon, 23 Jul 2007 10:29:13 -0700
Subject: A006336 - Unexpected Relation to Golden Ratio?
In-Reply-To: <20070722.072158.944.1.pauldhanna@juno.com>
References: <20070722.072158.944.1.pauldhanna@juno.com>
Message-ID: <d3dac270707231029l69ce27b4kb5601a6491a064cb@mail.gmail.com>

On 7/22/07, Paul D. Hanna <pauldhanna at juno.com> wrote:
>
>
> Seqfans,
>      Consider the nice sequence A006336:
> a(n) = a(n-1) + a(n-1 - number of even terms so far).
> http://www.research.att.com/~njas/sequences/A006336
> begins:
> [1,2,3,5,8,11,16,21,29,40,51,67,88,109,138,167,207,258,309,376,...].
>
> My COMMENT (NOT submitted to OEIS):
> -----------------------------------------------------------
> It seems that A006336 can be generated by a rule using the golden ratio:
>
> a(n) = a(n-1) + a([n/Phi]) for n>1 with a(1)=1  where Phi = (sqrt(5)+1)/2,
>
>
> i.e., the number of even terms up to position n-1 equals:
> n-1 - [n/Phi] for n>1 where Phi = (sqrt(5)+1)/2.

To simplify notation, let p = Phi = (sqrt(5)+1)/2.

Lemma. The sets { [n*p] : n=1,2,3,... } and { [n*p^2] : n=1,2,3,... }
are disjoint, and every positive integer belongs to one (and only
one!) of these sets.
I leave the proof of this Lemma to the reader as a challenge.

Theorem. The number of even terms in A006336 up to position n-1 equals
n-1 - [n/p].

Proof by induction:

Suppose that the number of even terms in A006336(1..n) equal n -
[(n+1)/p] for every n=2..m. In other words, A006336(n) is even iff n -
[(n+1)/p] = (n-1 - [n/p])  + 1, that is equivalent to:
A006336(n) == [(n+1)/p] - [n/p]  (mod 2).

We will prove that the same statement is true for n=m+1.

By the definition of A006336 and our induction hypothesis, we have
a(m+1) = a(m) + a([(m+1)/p]) == [(m+1)/p] - [m/p] + [([(m+1)/p]+1)/p]
- [[(m+1)/p]/p] (mod 2).
Therefore, we need to prove that
[(m+2)/p] - [(m+1)/p] == [(m+1)/p] - [m/p] + [([(m+1)/p]+1)/p] -
[[(m+1)/p]/p] (mod 2)
or
[(m+2)/p] + [m/p] + [([(m+1)/p]+1)/p] + [[(m+1)/p]/p] == 0 (mod 2).

Let m+1 = q*p + r, where q is integer and 0 < r < p, and q = s*p + t,
where s is integer and 0 < t < p. Then m+1 = (s*p + t)*p + r = s*p^2 +
t*p + r.

It is easy to see that [(m+2)/p] + [m/p] = 0 (mod 2) if and only if
p-1 < r < 1.
Similarly, [([(m+1)/p]+1)/p] + [[(m+1)/p]/p] == 0 (mod 2) if and only
if t < p-1.

There are three cases when [(m+2)/p] + [m/p] and [([(m+1)/p]+1)/p] +
[[(m+1)/p]/p] may have different oddness:

1) If p-1 < r < 1 and t > p-1 then m = [q*p] and m+1 = [(q+1)*p].
We also have
m+1 = s*p^2 + t*p + r > s*p^2 + (p-1)*p + p - 1 = (s+1)*p^2 - 1
and
m+1 = s*p^2 + t*p + r < s*p^2 + p*p + 1 = (s+1)*p^2 + 1,
implying that [(s+1)*p^2] = m+1 or [(s+1)*p^2] = m, a contradiction to Lemma.

2) If t < p-1 and r < p-1 then m = [q*p] and m+2 = [(q+1)*p].
We also have
m+1 = s*p^2 + t*p + r > s*p^2
and
m+1 = s*p^2 + t*p + r < s*p^2 + (p-1)*p + p-1 = (s+1)*p^2 - 1,
implying that either [s*p^2] = m or [(s+1)*p^2] = m+2, a contradiction to Lemma.

3) If t < p-1 and r > 1 then m-1 = [q*p], m+1 = [(q+1)*p].
We also have
m+1 = s*p^2 + t*p + r > s*p^2 + 1
and
m+1 = s*p^2 + t*p + r < s*p^2 + (p-1)*p + p = (s+1)*p^2
implying that either [s*p^2] = m-1 or [(s+1)*p^2] = m+1, a
contradiction to Lemma.

Therefore, [(m+2)/p] + [m/p] = 0 (mod 2) if and only if [(m+2)/p] +
[m/p] = 0 (mod 2), implying that [(m+2)/p] + [m/p] + [([(m+1)/p]+1)/p]
+ [[(m+1)/p]/p] == 0 (mod 2).

Q.E.D.

Max




From davidwwilson at comcast.net  Mon Jul 23 19:56:08 2007
From: davidwwilson at comcast.net (David Wilson)
Date: Mon, 23 Jul 2007 13:56:08 -0400
Subject: Divisor d is the total number of divisors
References: <F05775148D000C4B9321A5113D53CB7D405F91@KNTV-SERVER01.kntv.local>
Message-ID: <002c01c7cd52$bf849ce0$6501a8c0@yourxhtr8hvc4p>

These are the refactorable numbers, A033950.

----- Original Message ----- 
From: "Eric Angelini" <Eric.Angelini at kntv.be>
To: <seqfan at ext.jussieu.fr>
Sent: Monday, July 23, 2007 12:25 PM
Subject: Divisor d is the total number of divisors


>
>
> Hello SeqFans,
>
> Is this seq of interest?
> If yes could someone check and compute a few more terms?
>
> 1,8,9,12,18,24,36,...
>
> Integers I having one divisor which is also the total number of divisors 
> of I.
>
> 1 has 1 divisor which is 1
> 8 has 4 divs and 4 is one of them
> 9 has 3 divs and 3 is one of them
> 12 has 6 divs and 6 is one of them
> 18 has 6 divs and 6 is one of them
> 24 has 8 divs and 8 is one of them
> 36 has 9 divs and 9 is one of them
> ...
>
> 30 is not a member because 30 has 8 divs but not 8 itself : 
> [1,2,3,5,6,10,15,30]
>
> Best,
> ?.
>
>
>
>
>
>
> -- 
> No virus found in this incoming message.
> Checked by AVG Free Edition.
> Version: 7.5.476 / Virus Database: 269.10.14/912 - Release Date: 7/22/2007 
> 7:02 PM
> 




At 10:40 AM -0700 7/23/07, Max Alekseyev wrote:
>On 7/22/07, Paul D. Hanna <pauldhanna at juno.com> wrote:
>>
>>
>> Seqfans,
>>      Consider the nice sequence A006336:
>> a(n) = a(n-1) + a(n-1 - number of even terms so far).
>> http://www.research.att.com/~njas/sequences/A006336
>> begins:
>> [1,2,3,5,8,11,16,21,29,40,51,67,88,109,138,167,207,258,309,376,...].
>>
>> My COMMENT (NOT submitted to OEIS):
>> -----------------------------------------------------------
>> It seems that A006336 can be generated by a rule using the golden ratio:
>>
>> a(n) = a(n-1) + a([n/Phi]) for n>1 with a(1)=1  where Phi = (sqrt(5)+1)/2,
>>
>>
>> i.e., the number of even terms up to position n-1 equals:
>> n-1 - [n/Phi] for n>1 where Phi = (sqrt(5)+1)/2.
>
>To simplify notation, let p = Phi = (sqrt(5)+1)/2.

Nice, Max.

What does this sequence count? It is similar to A00123 and A005704, which
both have a recursion a(n)=a(n-1)+a([n/k]), where k is 2 and 3,
respectively.  Those sequences count "number of partitions of k*n into
powers of k". For sequence A006336, k=Phi.  Does A006336(n) count the
number of partitions of n*Phi into powers of Phi?

Tony





From noe at sspectra.com  Mon Jul 23 20:39:32 2007
From: noe at sspectra.com (T. D. Noe)
Date: Mon, 23 Jul 2007 11:39:32 -0700
Subject: A006336 - Unexpected Relation to Golden Ratio?
In-Reply-To: <d3dac270707231029l69ce27b4kb5601a6491a064cb@mail.gmail.com>
References: <20070722.072158.944.1.pauldhanna@juno.com>
 <d3dac270707231029l69ce27b4kb5601a6491a064cb@mail.gmail.com>
Message-ID: <p06240804c2caa4c86185@[192.168.1.43]>

Dear seqfans,

I#ve been surprised not to find sequences of the following form in the OEIS:

a(n)=min(k in N: sigma(r,n)=sigma(r,k)) with sigma(r,n)=sum of the r-th
power of the divisors of n:

new[r_, n_] :=
  (If[Head[#1] === rep, #1 = n, #1] & )[rep[DivisorSigma[r, n]]]

for r=0 (number of divisors)

Clear[rep];
(new[0, #1] & ) /@ Range[0, 100]

{0, 1, 2, 2, 4, 2, 6, 2, 6, 4, 6, 2, 12, 2, 6, 6, 16, 2, 12, 2, 12,
  6, 6, 2, 24, 4, 6, 6, 12, 2, 24, 2, 12, 6, 6, 6, 36, 2, 6, 6, 24,
  2, 24, 2, 12, 12, 6, 2, 48, 4, 12, 6, 12, 2, 24, 6, 24, 6, 6, 2,
  60, 2, 6, 12, 64, 6, 24, 2, 12, 6, 24, 2, 60, 2, 6, 12, 12, 6,
  24, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24, 2, 60, 6, 12, 6, 6, 6, 60,
  2, 12, 12, 36}

for r=1 (sum of divisors)

Clear[rep];
(new[1, #1] & ) /@ Range[0, 100]

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 6, 12, 13, 14, 14, 16, 10, 18,
  19, 20, 21, 22, 14, 24, 16, 20, 27, 28, 29, 30, 21, 32, 33, 34,
  33, 36, 37, 24, 28, 40, 20, 42, 43, 44, 45, 30, 33, 48, 49, 50,
  30, 52, 34, 54, 30, 54, 57, 40, 24, 60, 61, 42, 63, 64, 44, 66,
  67, 68, 42, 66, 30, 72, 73, 74, 48, 76, 42, 60, 57, 80, 81, 68,
  44, 84, 85, 86, 54, 88, 40, 90, 91, 60, 93, 66, 54, 96, 52, 98,
  99, 100}

r=2 (sum of squares of divisors)

Clear[rep];
(new[2, #1] & ) /@ Range[0, 100]

{0, 1, 2, 3, 4, 5, 6, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
  19, 20, 21, 22, 23, 24, 25, 24, 27, 28, 29, 30, 31, 32, 33, 34,
  30, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 40, 48, 49, 50,
  51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,
  67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 66, 78, 79, 80, 81, 82,
  83, 84, 85, 86, 87, 88, 89, 90, 78, 92, 93, 94, 95, 96, 97, 98,
  99, 100}

and so on.

Are these of interest? And if so, up to which exponent r?

Peter



>What does this sequence count? It is similar to A00123 and A005704, which
>both have a recursion a(n)=a(n-1)+a([n/k]), where k is 2 and 3,
>respectively.  Those sequences count "number of partitions of k*n into
>powers of k". For sequence A006336, k=Phi.  Does A006336(n) count the
>number of partitions of n*Phi into powers of Phi?

Answering my own question.

If the recursion starts with a(0)=1, then I think we obtain "number of
partitions of n*Phi into powers of Phi".  That sequence is 1, 2, 4, 6, 10,
16, 22,..., which I just submitted as A131882.

We need negative powers of Phi also, letting p=Phi and q=1/Phi

n=0:	0*p = {}				for 1 partition

n=1:	1*p = p = 1+q				for 2 partitions

n=2:	2*p = p+p = 1+p+q = 1+1+q+q = p^2+q	for 4 partitions

etc. tedious!

So A006336(n), which starts with a(1)=1, counts 1/2 of the "number of
partitions of n*Phi into powers of Phi"

Tony




From noe at sspectra.com  Mon Jul 23 21:39:53 2007
From: noe at sspectra.com (T. D. Noe)
Date: Mon, 23 Jul 2007 12:39:53 -0700
Subject: A006336 - Unexpected Relation to Golden Ratio?
In-Reply-To: <p06240804c2caa4c86185@[192.168.1.43]>
References: <20070722.072158.944.1.pauldhanna@juno.com>
 <d3dac270707231029l69ce27b4kb5601a6491a064cb@mail.gmail.com>
 <p06240804c2caa4c86185@[192.168.1.43]>
Message-ID: <p06240806c2cab0c23025@[192.168.1.43]>

------=_Part_156860_32560920.1185232257325
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Content-Transfer-Encoding: 7bit
Content-Disposition: inline

concerning the "duplicates"

http://www.research.att.com/~njas/sequences/?q=id:A087186|id:A087189

I think all terms of the second one are wrong.

A087189(1) should be = (A005191(3)-1)/2/3^2 = 1
A087189(2) should be = (A005191(7)-1)/2/7^2 = 83
etc, instead of what is listed :
A087189 2,4,31,304,40044,500522,86094668,1167752848,225001039696,

Thus these are no duplicates at all. PARI code :

A087189(n) = local(p=prime(n+2-(n==1))); (A005191(p)-1)/2/p^2
A005191(n) = sum(k=0,(2*n)\5,binomial(n,k)*binomial(-n,2*n-5*k))

attached both sequences up to n=100 ; below the corrected entry for
the wrong one.

M.H.

%I A087189
%S A087189 1, 83, 16907, 279021, 89444018, 1695011087, 658067703933,
5768410509553937, 122108313460051500, 1226978854222034501448,
593538703869555995238872, 13175226571428140572866093,
6594852118968926152838341468, 76339779380132942579800334346403
%N A087189 (P(p)-1)/2/p^2 where p runs through the odd primes
different from 5 and P(k) is
               the k-th central pentanomial coefficient (A005191).
%o A087189 (PARI) A087189(n) = local(p=prime(n+2-(n==1)));
(A005191(p)-1)/2/p^2 \\ - M.F. Hasler
%Y A087189 Adjacent sequences: A087186 A087187 A087188 this_sequence
A087190 A087191 A087192
%Y A087189 Sequence in context: A018294 A051569 A087186 this_sequence
A051759 A051570 A082913
%K A087189 nonn
%O A087189 1,1
%A A087189 Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 19 2003
%E A087189 Corrected and extended by Maximilian F. Hasler
(Maximilian.Hasler at gmail.com), Jul 23 2007

------=_Part_156860_32560920.1185232257325
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Content-Transfer-Encoding: base64
X-Attachment-Id: f_f4hkqnyy
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Njg2MjQ1MjQ0Mjk3NzM4NzU0OTI1Nzk1MjAxNjU1OTE5NTcyNzU2ODkyMTQz
NDIzMjExMzIyMTE4NTYyOTU0MjAyNjM1MjM5MTcwNzg1MTY1MjQ1ODY3NzAw
NDg3OTExMTM4ODQ4NjQ1NDIzOTY2OTUzMzc4NDg1MzE0MDc1MTcyMTE4OTkx
MjgwNTM5ODkzNDM1OTkyMDM5NzAwNzEzNzg2ODAwMzQ2OTc2NDQyNzUzCjk4
IDI4Nzc0ODk2ODc1NTYyODkyODYzOTI1MTc2NDE2NTI5Mjk5MTU2OTI4NTYy
MTIyNTk5NzQ2NDY3MTgwOTI2MjY5MDI0MzIxNDMwNzU1NDYyOTU2OTA1Mzk4
OTg0NTY3MDI2MTM0MjczOTc0NjgwNDAzNzYwODcwNDkzNzE4MDU2NzgwMzUx
NDkyNTMyOTY3MTIxNDAzMTE3MzE5NDAzOTIxNDI1MzcyOTQxMjE2NDg5MDE4
NDM4MDY3NjU5NDQyNjA4Njg5OTA2MTA1NjY3NTY0Njc3MTg3NTQ0MDQyNjQy
ODYyMjc4ODc5MDg3OTMwNjkzMzgyNzY0NzcxNzQ2Njg5MjQ3ODM1NjUxODk0
MTE5OTg4NTU0OTQwMzQwMjU4NDY0MjI3OTQ1MDkyMTc3NzM2MDgyMzgxMjMw
MTg5OTg2OTkyODc5Mzk5OTcwNTgyNzk4NTA5OTc5MjYzOTM5OTg1MjI4Njk1
NzgwMTgyNDMwMTI3Cjk5IDQzNzM4MTE2ODM2MjM5ODYyMDU0MjM3MjQ2ODAz
Mjc5NDU4MjEzNTEyMjc3Mjc5NzQzNDc1MDY3MTcyNzUxMTAxMDY5MzA3Mjkx
NDc5NTkzNjQ3OTQ4Njc2NjQ5MzQzMDMyNjkzMDAzMzg0OTMzMzg0MzU4MzYy
MDkxNzg2MjY3MDU1Mzc0MzM1NTM2MDc4MDI1NDgxOTM5MTQ4NjU4NjI4NzQ0
MTE2OTQxMzU0NjI2ODQ0Nzc2MTI0ODk4NjI3MzQzMTIyMzAyODA1MTYzNTA0
MTczNzM2MDA1MTI4MDEwMjYxNzg2NDA1MDA0NTI1NzY4MTg2MDUwMTY5ODk5
MTMzNDc2Mzg5MDU5NjUwOTU3NDM4NjI3NjY2NDY1MDA0NjM2Njk4NjkzOTMx
MjE5OTg5OTY1MzE3Njc0MDk5ODg2Nzk3ODYzOTEyMzk2OTIyOTg5OTgzOTgx
MTIwODkyNDI5NDAxMTM1MDEwNDY1Nzg3ODA1ODA5MwoxMDAgNDA4MjE4NTgx
MjQxMTM5Njg4ODAyNzgyMzY1ODQ2MDE5Mzg5NDAyNTM4Njk4NTMxMzk0NjU4
ODU1Nzg4Mzc1MzMwMDQzNTA0NjI3MDMxNjg4Mjk4ODcyMTI0MDMxNzM2MTc2
ODk5NjMzMjg0Mjc5Nzc0MTc3MjM2OTMzNTYwMTYxMzkwMjQzNjc2MTM2Nzg5
MDE0MzM2MTkxMDc4NDk5OTk1Mjg1MDg4MTIzMDA3Mzc5NjcxMDk3NzM0ODgy
NDQ4ODMzNDI5Mzg2NzIwNTM2MTk1NzA2NzkzMzc0MjAyNzk0MTAxOTUxOTI4
MzExNTkzMTE5NjgyOTkzMjMyMDU3OTA4NTAwOTI0NTY2ODU4MDE4NTgwMDkw
MzY2MzUwMDI5NTE2MzAzMzUzMjA5MTIyOTkxNTg4NDMyMzUxMjkyNTQ4MjY0
MjQ1MjMzODExNzI0NzIyNDAxNTc3Njg4MDMzMzkwNTM0NzEyMzMyMzAwNjkw
NDg1ODA0NjU0NTgxMwo=

------=_Part_156860_32560920.1185232257325--




From maximilian.hasler at gmail.com  Tue Jul 24 01:37:12 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Mon, 23 Jul 2007 19:37:12 -0400
Subject: Outstanding duplicates
In-Reply-To: <46a5200f.17bb720a.66c6.ffffffabSMTPIN_ADDED@mx.google.com>
References: <46a5200f.17bb720a.66c6.ffffffabSMTPIN_ADDED@mx.google.com>
Message-ID: <3c3af2330707231637i456739e7u74d5b064e8c5135@mail.gmail.com>

These :

http://www.research.att.com/~njas/sequences/?q=id:A023237|id:A105434

can be moved to the section "STRAIGHTFORWARD DUPES":

A023237  Numbers n such that n and 10n + 1 both prime.  	
A105434  Primes which with a 1 appended stay prime.

obviously the same.

M.H.




From alec at mihailovs.com  Tue Jul 24 02:52:14 2007
From: alec at mihailovs.com (Alec Mihailovs)
Date: Mon, 23 Jul 2007 19:52:14 -0500
Subject: A006336 - Unexpected Relation to Golden Ratio?
In-Reply-To: <d3dac270707231029l69ce27b4kb5601a6491a064cb@mail.gmail.com>
References: <20070722.072158.944.1.pauldhanna@juno.com> <d3dac270707231029l69ce27b4kb5601a6491a064cb@mail.gmail.com>
Message-ID: <1846E28BE6ED4805A69C8A9D6954FEA6@AlecPC>

From: "Max Alekseyev" <maxale at gmail.com>
Sent: Monday, July 23, 2007 12:29 PM

> On 7/22/07, Paul D. Hanna <pauldhanna at juno.com> wrote:
>>      Consider the nice sequence A006336:
>> a(n) = a(n-1) + a(n-1 - number of even terms so far).
>> http://www.research.att.com/~njas/sequences/A006336
>> -----------------------------------------------------------
>> It seems that A006336 can be generated by a rule using the golden ratio:
>> a(n) = a(n-1) + a([n/Phi]) for n>1 with a(1)=1  where Phi = 
>> (sqrt(5)+1)/2,
>
> Lemma. The sets { [n*p] : n=1,2,3,... } and { [n*p^2] : n=1,2,3,... }
> are disjoint, and every positive integer belongs to one (and only
> one!) of these sets.
> I leave the proof of this Lemma to the reader as a challenge.
>
> Theorem. The number of even terms in A006336 up to position n-1 equals
> n-1 - [n/p].

Another proof is based on the fact that A006336 mod 2 is A005614.

The proof is rather simple, but I don't have time to write it in a simple 
way at the moment.

I hope that Max can write a simple version of it based on his proof of the 
Theorem above.

By the way, looking at that, and searching for the initial terms of A001950, 
I found that A090909 seems to be a duplicate of A001950. Again, I don't have 
much of free time at the moment and I live it to Max to prove that.

Alec 





From jvospost3 at gmail.com  Tue Jul 24 03:11:31 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Mon, 23 Jul 2007 18:11:31 -0700
Subject: definition of anti-divisor
In-Reply-To: <5542af940707231709n330ca0ffkc581bb34984b1154@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
	 <5542af940707221117t4e411e40o840c9228f2e9b7f8@mail.gmail.com>
	 <3c3af2330707231221y5690ec3cm473f38b5f2bc6adc@mail.gmail.com>
	 <5542af940707231709n330ca0ffkc581bb34984b1154@mail.gmail.com>
Message-ID: <5542af940707231811t4044fa14rcdc33c9a227a9562@mail.gmail.com>

Any work since 2001 on whether or not there is a 6th anti-prime?  How
far has this been searched?  Any proofs or disproofs as to finiteness
of A066466?

COMMENT FROM Jonathan Vos Post RE A066466

%I A066466
%S A066466 3, 4, 6, 96, 393216
%N A066466 Numbers having just one anti-divisor.
%C A066466 Jon Perry calls these anti-prime numbers, saying that these
are the only 5 known. This sequence is worth extending, if possible,
or proving finite.
%F A066466 A066272(a(n)) = 1.
%Y A066466 Cf. A066272.
%O A066466 1
%K A066466 ,more,nonn,
%A A066466 Jonathan Vos Post (jvospost2 at yahoo.com), Jul 23 2007




From alec at mihailovs.com  Tue Jul 24 03:16:57 2007
From: alec at mihailovs.com (Alec Mihailovs)
Date: Mon, 23 Jul 2007 20:16:57 -0500
Subject: A006336 - Unexpected Relation to Golden Ratio?
In-Reply-To: <1846E28BE6ED4805A69C8A9D6954FEA6@AlecPC>
References: <20070722.072158.944.1.pauldhanna@juno.com> <d3dac270707231029l69ce27b4kb5601a6491a064cb@mail.gmail.com> <1846E28BE6ED4805A69C8A9D6954FEA6@AlecPC>
Message-ID: <707C3179321F466FAC06D9EDA30EF672@AlecPC>

I forgot to mention that 

n-1- (number of even terms thus far) = number of odd terms thus far 

and 

number of odd terms = the sum of the terms 

in a 0-1 sequence. 

Alec




From maxale at gmail.com  Tue Jul 24 03:57:27 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Mon, 23 Jul 2007 18:57:27 -0700
Subject: definition of anti-divisor
In-Reply-To: <5542af940707231811t4044fa14rcdc33c9a227a9562@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
	 <5542af940707221117t4e411e40o840c9228f2e9b7f8@mail.gmail.com>
	 <3c3af2330707231221y5690ec3cm473f38b5f2bc6adc@mail.gmail.com>
	 <5542af940707231709n330ca0ffkc581bb34984b1154@mail.gmail.com>
	 <5542af940707231811t4044fa14rcdc33c9a227a9562@mail.gmail.com>
Message-ID: <d3dac270707231857y6dae1f1cq27367b56568301cb@mail.gmail.com>

Except element 4, the elements of A066466 have form 2^k*p where p is
odd prime and both 2^(k+1)*p-1, 2^(k+1)*p+1 are prime (i.e., twin
primes).
In other words, A066466 without element 4 is a subsequence of A040040,
containing elements of the form 2^k*p with prime p.

Max

On 7/23/07, Jonathan Post <jvospost3 at gmail.com> wrote:
> Any work since 2001 on whether or not there is a 6th anti-prime?  How
> far has this been searched?  Any proofs or disproofs as to finiteness
> of A066466?
>
> COMMENT FROM Jonathan Vos Post RE A066466
>
> %I A066466
> %S A066466 3, 4, 6, 96, 393216
> %N A066466 Numbers having just one anti-divisor.
> %C A066466 Jon Perry calls these anti-prime numbers, saying that these
> are the only 5 known. This sequence is worth extending, if possible,
> or proving finite.
> %F A066466 A066272(a(n)) = 1.
> %Y A066466 Cf. A066272.
> %O A066466 1
> %K A066466 ,more,nonn,
> %A A066466 Jonathan Vos Post (jvospost2 at yahoo.com), Jul 23 2007
>




From maxale at gmail.com  Tue Jul 24 04:29:49 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Mon, 23 Jul 2007 19:29:49 -0700
Subject: definition of anti-divisor
In-Reply-To: <d3dac270707231857y6dae1f1cq27367b56568301cb@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
	 <5542af940707221117t4e411e40o840c9228f2e9b7f8@mail.gmail.com>
	 <3c3af2330707231221y5690ec3cm473f38b5f2bc6adc@mail.gmail.com>
	 <5542af940707231709n330ca0ffkc581bb34984b1154@mail.gmail.com>
	 <5542af940707231811t4044fa14rcdc33c9a227a9562@mail.gmail.com>
	 <d3dac270707231857y6dae1f1cq27367b56568301cb@mail.gmail.com>
Message-ID: <d3dac270707231929r558496f6t36dfb40745bb02fe@mail.gmail.com>

Furthermore, since 2^(k+1)*p-1, 2^(k+1)*p+1 must equal -1 and +1
modulo 3, the number 2^(k+1)*p must be 0 modulo 3, implying that p=3.

Therefore, every element of A066466, except 4, must be of the form
3*2^k such that 3*2^(k+1)-1, 3*2^(k+1)+1 are twin primes. There no
such new k (i.e., except known 1,2,6,18) below 1000.

Max

On 7/23/07, Max Alekseyev <maxale at gmail.com> wrote:
> Except element 4, the elements of A066466 have form 2^k*p where p is
> odd prime and both 2^(k+1)*p-1, 2^(k+1)*p+1 are prime (i.e., twin
> primes).
> In other words, A066466 without element 4 is a subsequence of A040040,
> containing elements of the form 2^k*p with prime p.
>
> Max
>
> On 7/23/07, Jonathan Post <jvospost3 at gmail.com> wrote:
> > Any work since 2001 on whether or not there is a 6th anti-prime?  How
> > far has this been searched?  Any proofs or disproofs as to finiteness
> > of A066466?
> >
> > COMMENT FROM Jonathan Vos Post RE A066466
> >
> > %I A066466
> > %S A066466 3, 4, 6, 96, 393216
> > %N A066466 Numbers having just one anti-divisor.
> > %C A066466 Jon Perry calls these anti-prime numbers, saying that these
> > are the only 5 known. This sequence is worth extending, if possible,
> > or proving finite.
> > %F A066466 A066272(a(n)) = 1.
> > %Y A066466 Cf. A066272.
> > %O A066466 1
> > %K A066466 ,more,nonn,
> > %A A066466 Jonathan Vos Post (jvospost2 at yahoo.com), Jul 23 2007
> >
>




From maxale at gmail.com  Tue Jul 24 07:02:50 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Mon, 23 Jul 2007 22:02:50 -0700
Subject: definition of anti-divisor
In-Reply-To: <d3dac270707231929r558496f6t36dfb40745bb02fe@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
	 <5542af940707221117t4e411e40o840c9228f2e9b7f8@mail.gmail.com>
	 <3c3af2330707231221y5690ec3cm473f38b5f2bc6adc@mail.gmail.com>
	 <5542af940707231709n330ca0ffkc581bb34984b1154@mail.gmail.com>
	 <5542af940707231811t4044fa14rcdc33c9a227a9562@mail.gmail.com>
	 <d3dac270707231857y6dae1f1cq27367b56568301cb@mail.gmail.com>
	 <d3dac270707231929r558496f6t36dfb40745bb02fe@mail.gmail.com>
Message-ID: <d3dac270707232202q7d6d4198t259627c7b9977368@mail.gmail.com>

On 7/23/07, Max Alekseyev <maxale at gmail.com> wrote:

> Therefore, every element of A066466, except 4, must be of the form
> 3*2^k such that 3*2^(k+1)-1, 3*2^(k+1)+1 are twin primes. There no
> such new k (i.e., except known 1,2,6,18) below 1000.

Small corrections: 1,2,6,18 above correspond to k+1 not k.
In other words, 3*2^n - 1, 3*2^n + 1 are twin primes for n=1,2,6,18.

According to these tables:
http://www.prothsearch.net/riesel.html
http://www.prothsearch.net/riesel2.html
there are no other such n up to 1200000.
Therefore, the next element of A066466 (if it exists) is greater than
3*2^1200000 ~= 10^361236.

Max




From jeremy.gardiner at btinternet.com  Tue Jul 24 11:35:35 2007
From: jeremy.gardiner at btinternet.com (JEREMY GARDINER)
Date: Tue, 24 Jul 2007 10:35:35 +0100 (BST)
Subject: early bird sequence
In-Reply-To: <d3dac270707232202q7d6d4198t259627c7b9977368@mail.gmail.com>
Message-ID: <836624.76115.qm@web86605.mail.ukl.yahoo.com>

Cf.  A048991 and A048992 (Rollman numbers)
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From maximilian.hasler at gmail.com  Tue Jul 24 21:52:51 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Tue, 24 Jul 2007 15:52:51 -0400
Subject: early bird sequence
In-Reply-To: <836624.76115.qm@web86605.mail.ukl.yahoo.com>
References: <d3dac270707232202q7d6d4198t259627c7b9977368@mail.gmail.com>
	 <836624.76115.qm@web86605.mail.ukl.yahoo.com>
Message-ID: <3c3af2330707241252g52e73c0awb1ad943f7562cb7a@mail.gmail.com>

thanks for the reference.

someone could add an explanation to either of those to explain why
they are not the same (e.g. 12 being omitted, 21 is no more "earlier
in the sequence").

M.H.


On 7/24/07, JEREMY GARDINER <jeremy.gardiner at btinternet.com> wrote:
> Cf.  A048991 and A048992 (Rollman numbers)




From maxale at gmail.com  Wed Jul 25 05:30:36 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Tue, 24 Jul 2007 20:30:36 -0700
Subject: stapled intervals (following A090318)
Message-ID: <d3dac270707242030rdf5113wa94c3bd79ed6c689@mail.gmail.com>

SeqFans,

I'm about to submit the following sequences you may find entertaining.

A090318 defines stapled sequence as an interval of positive integers
that does not contain an element coprime to every other element of the
interval. In other word, a sequence is stapled if for every element x
there is another element y (different from x) such that gcd(x,y)>1.

The shortest stapled interval has length 17 and starts with the number 2184.
A090318 gives the smallest stapled interval of the given length n>=17.

In particular, it is interesting to notice that the intervals
[27829,27846] and [27828,27846] are stapled while the interval
[27828,27845] is not.

It is clear that a stapled interval [a,b] may not contain a prime
number greater than b/2 (as such a prime would be coprime to every
other element of the interval). Together with Bertrand's Postulate
that implies a>b/2 or b<2a. And it follows that
* a stapled interval may not contain prime numbers at all;
* for any particular positive integer a, we can determine if it is a
starting point of some stapled interval.

Sequence of starting points of stapled intervals is:
2184, 27828, 27829, 27830, 32214, 57860, 62244, 87890, 92274, 110990,
117920, 122304, 127374, 147950, 151058, 151059, 151060, 151061,
151062, 152334, 163488, 171054, 177980, 182364, 185924, 185925,
185926, 208010, 212394, 238040, 242424, 249678, 260810, 260811,
260812, 260813, 260814, 264498, 268070, 272454, 298100, 302484,
320870, 320871, 320872, 323510, 324564, 328130, 332514, 339434,
339435, 339436, 339437, 339438, 347004, 358160, 362544, 388190,
392574, 399500, 409188, 409189, 409190, 418220, 422600, 422601,
422602, 422603, 422604, 448250, 452634, 471014, 471015, 471016,
478280, 482664

Call a stapled interval "maximum" if it is not a proper sub-interval
of any other stapled interval. Starting points of maximum stapled
intervals are:
2184, 27828, 32214, 57860, 62244, 87890, 92274, 110990, 117920,
122304, 127374, 147950, 151058, 152334, 163488, 171054, 177980,
182364, 185924, 208010, 212394, 238040, 242424, 249678, 260810,
264498, 268070, 272454, 298100, 302484, 320870,  323510, 324564,
328130, 332514, 339434, 347004, 358160, 362544, 388190, 392574,
399500, 409188, 418220, 422600, 448250, 452634, 471014, 478280, 482664

Similarly, call a stapled interval "minimum" if it does not contain
any stapled proper subinterval. Starting points of minimum stapled
intervals are:
2184, 27830, 32214, 57860, 62244, 87890, 92274, 110990, 117920,
122304, 127374, 147950, 151062, 152334, 163488, 171054, 177980,
182364, 185926, 208010, 212394, 238040, 242424, 249678, 260814,
264498, 268070, 272454, 298100, 302484, 320872, 323510, 324564,
328130, 332514, 339438, 347004, 358160, 362544, 388190, 392574,
399500, 409190, 418220, 422604, 448250, 452634, 471016, 478280, 482664

Max




From maxale at gmail.com  Wed Jul 25 05:48:56 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Tue, 24 Jul 2007 20:48:56 -0700
Subject: full rank designs over Z_m
Message-ID: <d3dac270707242048t53cd7745ybdfbbfcee318053f@mail.gmail.com>

SeqFans,

Does the following sounds familiar to anybody?

Fix an integer m>=2. For any positive integer n consider a matrix k x
n over Z_m such that any n rows (out of k) are linearly independent.
Define a(n) as the maximum possible k.

 From practical point of view such construction gives rise to a number
of sequences for different m. Are they in OEIS?

I'm also interested in the theory behind. I believe such matrices
should have been studied.

Thanks,
Max




From bdm at cs.anu.edu.au  Wed Jul 25 06:23:33 2007
From: bdm at cs.anu.edu.au (Brendan McKay)
Date: Wed, 25 Jul 2007 14:23:33 +1000
Subject: full rank designs over Z_m
In-Reply-To: <d3dac270707242048t53cd7745ybdfbbfcee318053f@mail.gmail.com>
References: <d3dac270707242048t53cd7745ybdfbbfcee318053f@mail.gmail.com>
Message-ID: <20070725042333.GA8211@cs.anu.edu.au>


In the case of m not prime, one has to be cautious because several
equivalences from the prime case do not hold.  In particular,
being linearly independent is different from having a full-sized
submatrix (in your case, n*n) of non-zero determinant.

I wrote a paper long ago with Richard Brent that counts integer
matrices with given rank defined by the submatrix-determinant method:
   http://cs.anu.edu.au/~bdm/papers/BrentMcKayZm.pdf
I don't see how it applies to your problem easily, but maybe some
of the ideas are useful.

Also, regardless of your problem, that paper seems to provide any
number of sequences not in OEIS (so this is an invitation for some
enterprising oeiser to jump in).  The paper refers to probabilities
but of course you just need to multiply by the total number of
matrices to turn it into a count.

Brendan.


* Max Alekseyev <maxale at gmail.com> [070725 13:49]:
> SeqFans,
> 
> Does the following sounds familiar to anybody?
> 
> Fix an integer m>=2. For any positive integer n consider a matrix k x
> n over Z_m such that any n rows (out of k) are linearly independent.
> Define a(n) as the maximum possible k.
> 
> >From practical point of view such construction gives rise to a number
> of sequences for different m. Are they in OEIS?
> 
> I'm also interested in the theory behind. I believe such matrices
> should have been studied.
> 
> Thanks,
> Max




From jvospost3 at gmail.com  Wed Jul 25 06:26:38 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Tue, 24 Jul 2007 21:26:38 -0700
Subject: definition of anti-divisor
In-Reply-To: <d3dac270707232202q7d6d4198t259627c7b9977368@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
	 <5542af940707221117t4e411e40o840c9228f2e9b7f8@mail.gmail.com>
	 <3c3af2330707231221y5690ec3cm473f38b5f2bc6adc@mail.gmail.com>
	 <5542af940707231709n330ca0ffkc581bb34984b1154@mail.gmail.com>
	 <5542af940707231811t4044fa14rcdc33c9a227a9562@mail.gmail.com>
	 <d3dac270707231857y6dae1f1cq27367b56568301cb@mail.gmail.com>
	 <d3dac270707231929r558496f6t36dfb40745bb02fe@mail.gmail.com>
	 <d3dac270707232202q7d6d4198t259627c7b9977368@mail.gmail.com>
Message-ID: <5542af940707242126q68201c55pf5c674ce6848194f@mail.gmail.com>

NEW SEQUENCE FROM Jonathan Vos Post

 %I A000001
 %S A000001 2, 6, 36, 144, 4320, 64800, 777600,
 65318400, 2743372800
 %N A000001 Anti-divisorial; the product of all anti-divisors of all
integers equal or less than n.
 %C A000001 Different from the anti-primorial, which is the partial
products of anti-primes.
 %F A000001 a(n) = PRODUCT[k=3..n] {anti-divisors(k)}
 = PRODUCT[k=3..n] PRODUCT[j=1..A066272(k)] (j-th
 element of k-th row of A130799) = partial products of A091507.
 %e A000001 a(11) = anti-divisors of 3 * anti-divisors of 4 * ... *
anti-divisors of 11 = (2) * (3) * (2 * 3) * (4) * (2 * 3 * 5) * (3 *
5) * (2 * 6) * (3 * 4 * 7) * (2 * 3 * 7) = 2743372800.
 %Y A000001 Cf. A066272, A091507, A130799.
 %O A000001 3,1
 %K A000001 ,easy,nonn,
 %A A000001 Jonathan Vos Post (jvospost2 at yahoo.com), Jul 25 2007
 RH
 RA 192.20.225.32




From maxale at gmail.com  Wed Jul 25 11:49:44 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Wed, 25 Jul 2007 02:49:44 -0700
Subject: ps Re Divisors concatenated shape a prime
In-Reply-To: <33a322bc0707200443l833a591gc4abfcff9e172a43@mail.gmail.com>
References: <200707201002.l6KA2ZKE432256@fry.research.att.com>
	 <d3dac270707200436x7996f230qdeda7d1a243a730e@mail.gmail.com>
	 <33a322bc0707200443l833a591gc4abfcff9e172a43@mail.gmail.com>
Message-ID: <d3dac270707250249u1df96051te33cf39151d8ce1e@mail.gmail.com>

Three more prime divisors have been found with ECM:
1303590319,
2029994921,
and
2635016943923513073981336313737132619.

The co-factor is still composite and requires further factorization.

Regards,
Max

On 7/20/07, Simon Plouffe <simon.plouffe at gmail.com> wrote:
> The number is
> 23637741111132222263396784181836212543250863192525397638\
>     5050794957757619119155152382118123439689236246879378\
>     3543703190673607553698617087406381347215107397221082\
>     2661095832164532219166133479486848572669589736971440\
>     0438460545718008769210914230132785470246166539693602\
>     6557094049233307938690398356410738499619079180796712\
>     8214769992381581114913062399108161968641222982612479\
>     8216323937282334473918719732448590592334050047581378\
>     1681898530966894783743946489718118466810009516275633\
>     6379706181021501427441345045695592720430028548826900\
>     9139118541259851760510992223024564332519703521021984\
>     4460491286637795552815329766690736929975591105630659\
>     5333814738598961996828961134745198615110831211923993\
>     6579222694903972302216624228859904868834042355958453\
>     3249363577198097376680847119169066498726355938826715\
>     6198557234875910075477711877653431239711446975182015\
>     0954106781648014685956717046277302264311087056416726\
>     0822620744350752392673213563296029371913434092554604\
>     5286221741128334521645241488701504785346326116925017\
>     8246786223305225717801965223385003564935724466104514\
>     3560384022108741886504369675409778385289018044217483\
>     7730087393508195567705780212066326225659513109026229\
>     33515586703
>
> and divisible by 13 and 47 if I am not mistaking.
>
> the rest (%/13/47) is not prime either but I can't determine
> its factors.
>
> simon plouffe
>




From zakseidov at yahoo.com  Wed Jul 25 15:42:55 2007
From: zakseidov at yahoo.com (zak seidov)
Date: Wed, 25 Jul 2007 06:42:55 -0700 (PDT)
Subject: A005589: Number of letters in the English name of n, excluding spaces and hyphens
Message-ID: <320334.92018.qm@web38204.mail.mud.yahoo.com>

Dear seqfans,

I'm going to submit 
b005589.txt and a005589.txt
(after end of OEIS vacation?).

In the process, I've found that 
numbers 800 and 900 (with their English names)
are missed in G. Schildberger's file
http://www.research.att.com/~njas/sequences/a000027.txt


If someone's interested I can send both files - 
for checking/extending.  

Thanks, 
Zak

 
A005589  Number of letters in the English name of n,
excluding spaces and hyphens.  
%%%%%%%%%%%

b005589.txt 
0 4
1 3
2 3
3 5
4 4
<skip>
1017 20
1018 20
1019 19
1020 17
1021 20
1022 20
%%%%%%%%%%%%

a005589.txt
zero
one
two
three
four
<skip>
onethousandseventeen
onethousandeightteen
onethousandnineteen
onethousandtwenty
onethousandtwentyone
onethousandtwentytwo


       
____________________________________________________________________________________
Got a little couch potato? 
Check out fun summer activities for kids.
http://search.yahoo.com/search?fr=oni_on_mail&p=summer+activities+for+kids&cs=bz 




I will be doing occasional updates, but that's all.

Neil




From njas at research.att.com  Wed Jul 25 18:51:08 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Wed, 25 Jul 2007 12:51:08 -0400 (EDT)
Subject: The OEIS "vacation" continues through the end of Sept.
In-Reply-To: <320334.92018.qm@web38204.mail.mud.yahoo.com>
References: <320334.92018.qm@web38204.mail.mud.yahoo.com>
Message-ID: <200707251651.l6PGp8YP2000882@fry.research.att.com>

Zak is right.  I'd found that a year ago, when I built a table of the
alphabetical reversal of the standard names of nonnegative integers
from G. Schildberger's draft.

He also had one or two misspellings, as I recall.

On 7/25/07, zak seidov <zakseidov at yahoo.com> wrote:
> Dear seqfans,
>
> I'm going to submit
> b005589.txt and a005589.txt
> (after end of OEIS vacation?).
>
> In the process, I've found that
> numbers 800 and 900 (with their English names)
> are missed in G. Schildberger's file
> http://www.research.att.com/~njas/sequences/a000027.txt
>
>
> If someone's interested I can send both files -
> for checking/extending.
>
> Thanks,
> Zak
>
>
> A005589  Number of letters in the English name of n,
> excluding spaces and hyphens.
> %%%%%%%%%%%
>
> b005589.txt
> 0 4
> 1 3
> 2 3
> 3 5
> 4 4
> <skip>
> 1017 20
> 1018 20
> 1019 19
> 1020 17
> 1021 20
> 1022 20
> %%%%%%%%%%%%
>
> a005589.txt
> zero
> one
> two
> three
> four
> <skip>
> onethousandseventeen
> onethousandeightteen
> onethousandnineteen
> onethousandtwenty
> onethousandtwentyone
> onethousandtwentytwo
>
>
>
> ____________________________________________________________________________________
> Got a little couch potato?
> Check out fun summer activities for kids.
> http://search.yahoo.com/search?fr=oni_on_mail&p=summer+activities+for+kids&cs=bz
>




From petsie at dordos.net  Thu Jul 26 02:36:09 2007
From: petsie at dordos.net (Peter Pein)
Date: Thu, 26 Jul 2007 02:36:09 +0200
Subject: Link to Mathematica script is outdated (says an advertising pest)
Message-ID: <46A7EC79.4000406@dordos.net>

The link to a Mma-script to easy put sequences in the OEIS-format on the
page http://www.research.att.com/~njas/sequences/Submit.html which
should link to a Mma-file http://www.seqfan.net/EISFormat.m leads me to
a strange (hijacking?) site full of advertising which states that
"seqfan.net expired on 07/12/2007 and is pending renewal or deletion."

If I remember correctly, there's a package at mathworld.wolfram.com with
some utilities needed for Eric's project which include some EISFormat
functionality.




From jvospost3 at gmail.com  Thu Jul 26 04:23:36 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Wed, 25 Jul 2007 19:23:36 -0700
Subject: A005589: Number of letters in the English name of n, excluding spaces and hyphens
In-Reply-To: <793412.80456.qm@web38202.mail.mud.yahoo.com>
References: <5542af940707251527x10cfb2c8sefbebd8546c11f59@mail.gmail.com>
	 <793412.80456.qm@web38202.mail.mud.yahoo.com>
Message-ID: <5542af940707251923u5e8e02c4ua885db0f05bceffe@mail.gmail.com>

Zak:

You are right. Same misspellings that I found.

===================

Source: Webster's Revised Unabridged Dictionary (1913)

Eighteen \Eight"een`\, a. [AS. eahtat?ne, eahtat?ne. See
   Eight, and Ten, and cf. Eighty.]
   Eight and ten; as, eighteen pounds.

Eighteen \Eight"een`\, n.
   1. The number greater by a unit than seventeen; eighteen
      units or objects.

   2. A symbol denoting eighteen units, as 18 or xviii.

===================

    * Is Amendment Eighteen Treason? by Joshua Grozier

Author(s) of Review: Charles Hall Davis
Virginia Law Review, Vol. 17, No. 2 (Dec., 1930), pp. 211-214
doi:10.2307/1066265
===================




From petsie at dordos.net  Thu Jul 26 05:19:41 2007
From: petsie at dordos.net (Peter Pein)
Date: Thu, 26 Jul 2007 05:19:41 +0200
Subject: a propos divisors...
In-Reply-To: <46A50268.3040307@dordos.net>
References: <46A50268.3040307@dordos.net>
Message-ID: <46A812CD.60303@dordos.net>

Peter Pein schrieb:
> Dear seqfans,
> 
> I#ve been surprised not to find sequences of the following form in the OEIS:
> 
> a(n)=min(k in N: sigma(r,n)=sigma(r,k)) with sigma(r,n)=sum of the r-th
> power of the divisors of n:
...
> Are these of interest? And if so, up to which exponent r?
> 
> Peter
> 

Although the interst seems to be bounded by a fairly small n (0 until
now), I would like to ask you if you could please help me.

I decided to publish the sequences in two forms:

a) n such there is at least one x<n such that sigma(r,x)=sigma(r,n),
because there exists such a series for r=1 (A069822)

and

b) a(n)=min({k>0: sigma(r,k)=sigma(r,n)})


The issue with these is that they behave in a surprising way for r=3 (I
think, sequences for higher exponents are too hard to calculate without
 tricks)

for r = 0 the type-a-seq starts:
3, 5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20
for r = 1:
11, 15, 17, 23, 25, 26, 31, 35, 38, 39, 41, 46, 47, 51
for r = 2
7, 26, 35, 47, 77, 91, 119, 130, 133, 141, 157, 161, 175

but for r = 3 the air becomes very thin. If I did not make a mistake,
this seq starts(!):

194315, 295301, 590602, 1181204, 1476505, 1886920, 2067107, 2362408,
2526095, 2953010, 3248311, 3691985, 3838913, 4134214, 4469245, 4724816,
5020117, 5610719, 5635135, 5906020

If any of you with some Mathematica knowledge could please confirm, that
the following lines calculate the sequence described above?

Timing[Block[{spa, nmax = 6*10^6, expo = 3},
   Reap[For[n = 1, n <= nmax, n++,
      (If[Head[#1] === spa, #1 = n, Sow[{n, #1}]] & )[
       spa[DivisorSigma[expo, n]]]]][[2,1]]]]

the name spa is an artefact; I tried this with SparseArrays, but the
allowed range of indices has not been sufficient. I use it as an
initially undefined function (Block[{spa..}]). For each n to test I look
wether spa[sigma(r,n)] has been defined. If not, the Head is still spa
and I set spa[sigma(r,n)] to n; else the remembered value together with
n will go to the result via the Sow-Reap mechanism. This way I get
type-a and a hint for type-b sequences in one run of the proggie.

And if you want to make me really happy (I have had birthday on July
24th ;-) (http://www.stevesbeatles.com/songs/when_im_sixty_four.asp this
age will be reached in 20 years, but the symptoms... )):

 If you've got Mathematica and more RAM (4GB or so) than I do (1.5 GB),
could you please run this code  with, say nmax=10 or 20 million? On my
machine it swapped heavily with nmax=6 million and I had to kill
MathKernel as I tried nmax=10^7. The lines above took ~181 seconds to
evaluate (nmax=10^7 has been stopped by me after 15 minutes). I do not
expect any runtimes of more than 7 minutes. Would this be possible, please?

 Alternatively any hints how to calculate these sequences more efficient
would be highly appreciated (AFAIK there exists no kind of "inverse
function" to sigma(r,n) w.r.t. n which could be calculated without this
brute-force method).

Thank you for your attention and in advance for CPU-time,

Peter



* Peter Pein <petsie at dordos.net> [Jul 26. 2007 08:51]:
> The link to a Mma-script to easy put sequences in the OEIS-format on the
> page http://www.research.att.com/~njas/sequences/Submit.html which
> should link to a Mma-file http://www.seqfan.net/EISFormat.m leads me to
> a strange (hijacking?) site full of advertising which states that
> "seqfan.net expired on 07/12/2007 and is pending renewal or deletion."

This is not a domain grabber but "network solutions"

> 
> If I remember correctly, there's a package at mathworld.wolfram.com with
> some utilities needed for Eric's project which include some EISFormat
> functionality.

http://www.aboutus.org/Seqfan.net
ogerard at ext.jussieu.fr (CC)

The domain should simply be renewed.
Else at some point in time a domain grabber will get it.




From arndt at jjj.de  Thu Jul 26 10:50:45 2007
From: arndt at jjj.de (Joerg Arndt)
Date: Thu, 26 Jul 2007 10:50:45 +0200
Subject: Link to Mathematica script is outdated (says an advertising pest)
In-Reply-To: <46A7EC79.4000406@dordos.net>
References: <46A7EC79.4000406@dordos.net>
Message-ID: <20070726085045.GA8464@amd32.purzl.net>

On 6/9/06, Max <maxale at gmail.com> wrote:
> On 6/9/06, Graeme McRae <g_m at mcraefamily.com> wrote:
>
> > It took me quite some time to understand how Ranier Rosenthal is counting
> > rounds, before I obtained A112088 as the number of rounds necessary to kill
> > all but one of n players of the Josephus Game with every third man out.  I
> > first tried using Hugo Pfoertner's idea where the final survivor counts the
> > times he was passed by the executioner including the final rendezvous.  But
> > that method gives an entirely different sequence (A005428:
> > 1,2,3,4,6,9,14,21,31,47,70,105,158,...).
>
> I've also ended up with this exactly sequence in my computations.

I have found an old draft message where I was explaining the details
of my computation.
I reproduce them below in hope that someone will find them useful.

Suppose that we have x persons at the circle (enumerated from 0 to
x-1) and the executioner stands at the person number s. After one
round we will eliminate 1+[(x-s-1)/3] persons and will end up at the
person number (s-x) mod 3. This observation leads to the following
algorithm (stated in the form of PARI/GP program) counting the
required number of rounds:

{ a(n) = local(count = 0, s = 0, x = n, x_new);
while(x>1,
   x_new = x - 1 - (x-s-1)\3;
   s = (s-x) % 3;
   x = x_new;
   count++;
);
count
}

For n=1..50, PARI/GP gives:

? vector(50,n,a(n))
%1 = [0, 1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7,
7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9,
9, 9, 9, 9, 9, 9]

This sequence (counting the number of rounds) seems to be missing in OEIS.

Now we want to "reverse" this algorithm, i.e., starting with a single
person make the given number of rounds r and output n. This involves
solving of the following system of equations:

x_new = x_old - 1 - [(x_old - s_old - 1)/3];
s_new = (s_old - x_old) mod 3;

with respect to x_old and s_old.
It is useful to notice that
[(x_old - s_old - 1)/3] = (x_old - s_old + s_new)/3 - 1,
implying
x_new = x_old - (x_old - s_old + s_new)/3.

Therefore,

x_old = (3*x_new + s_new)/2 - s_old/2.

 From this equation we get value of s_old modulo 2 (it must make x_old
integer). However, that does not uniquely define s_old since it can
take 3 values: 0, 1, and 2. But we can follow greedy strategy here
(though I'm not going to prove that) making x_old the smallest
possible on each round. In other words, we can limit values of s_old
to 1 and 2, and state that

x_old = [(3*x_new + s_new - 1)/2].
s_old = (s_new + x_old) mod 3;

That leads to the following PARI/GP implementation:

{ b(n) = local( x=1, s=0 );
for(r=1,n,
  x = (3*x + s)\2;
  s = (s + x) % 3;
);
x
}

The first 50 values are:

? vector(50,n,b(n))
%1 = [1, 2, 3, 4, 6, 9, 14, 21, 31, 47, 70, 105, 158, 237, 355, 533,
799, 1199, 1798, 2697, 4046, 6069, 9103, 13655, 20482, 30723, 46085,
69127, 103691, 155536, 233304, 349956, 524934, 787401, 1181102,
1771653, 2657479, 3986219, 5979328, 8968992, 13453488, 20180232,
30270348, 45405522, 68108283, 102162425, 153243637, 229865456,
344798184, 517197276]

That is exactly the sequence A005428. Based on our method of
computing, we can give an alternative description of A005428:

Define a mapping f over vectors with two integer components as follows:
f: (x,s)   -->   ( [(3*x + s) / 2], (s + x) mod 3 )

Then A005428(n) is the first component of the n-th iteration of f on
the vector (1,0).

Regards,
Max




From r.rosenthal at web.de  Thu Jul 26 22:24:34 2007
From: r.rosenthal at web.de (Rainer Rosenthal)
Date: Thu, 26 Jul 2007 22:24:34 +0200
Subject: Sequence A112088 Motivation and Example?
In-Reply-To: <d3dac270707261223l69633b85n2277d5adfc2f5869@mail.gmail.com>
References: <4489D93F.3080902@web.de>	 <020401c68c0a$9470a650$6900000a@mcraeclan.com>	 <02a801c68c1c$1421f320$6900000a@mcraeclan.com>	 <d3dac270606092351x3c02aeeo84c20b6871eb7e68@mail.gmail.com> <d3dac270707261223l69633b85n2277d5adfc2f5869@mail.gmail.com>
Message-ID: <46A90302.8020905@web.de>

Max Alekseyev wrote:
> On 6/9/06, Max <maxale at gmail.com> wrote:
>>On 6/9/06, Graeme McRae <g_m at mcraefamily.com> wrote:
>>>It took me quite some time to understand how Ranier Rosenthal is counting
                                                Rainer (please)
>>>rounds, before I obtained A112088 as the number of rounds necessary to kill
>>>all but one of n players of the Josephus Game with every third man out.

Many many thanks for this post!

Some days ago I found my last scribblings, which were meant for
another comment to A112088. But it's all that long ago ... I
didn't understand anything of what I wrote :-(

I'm going to print your mail now and maybe I will find the
energy to get back to all these lovely round-countings.

As far as I remember I wasn't able to really grasp the idea of
the original submitter (Simon Strandgaard). From the discussions
in de.sci.mathematik and de.rec.denksport I remember that there
were some remarks giving hints to Knuth. Oops ... long time ago.

Friendly greetings,
Rainer






From petsie at dordos.net  Fri Jul 27 02:26:33 2007
From: petsie at dordos.net (Peter Pein)
Date: Fri, 27 Jul 2007 02:26:33 +0200
Subject: [Fwd: SEQ FROM Peter Pein]
Message-ID: <46A93BB9.1010209@dordos.net>

Dear Neil,

I obviously made a mistake, when entering the sequence number which I've
got from the dispenser. This should become A131903.

sorry,
Peter

The following is a copy of the email message that was sent to njas
containing the sequence you submitted.

All greater than and less than signs have been replaced by their html
equivalents.  They will be changed back when the message is processed.

This copy is just for your records.  No reply is expected.
 Subject: NEW SEQUENCE FROM Peter Pein


%I A000001
%S A000001 3, 5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22,
23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63
%N A000001 Those integers for which a smaller positive integer exists
which has the same number of divisors
%F A000001 a(n)= n-th element of {x&gt;0, there exists a k with
0&lt;k&lt;x and the same number of divisors as x)
%e A000001 a(4)=8 because it is the fourth integer for which a smaller
integer with the same number of divisors exists (after 3, 5 and 7).
divisors of 8 are 1,2,4,8 which are four and the divisors of 6 which is
less than 8 are (1, 2, 3, 6) which are four.
%t A000001 Clear[tmp];
Function[n, If[Head[#1] === tmp, #1 = n; Unevaluated[Sequence[]], n] &
   [tmp[DivisorSigma[0, n]]]] /@ Range[64]
%Y A000001 Cf. A069822, A131902-A131908
%O A000001 1
%K A000001 ,easy,nonn,
%A A000001 Peter Pein (petsie at dordos.net), Jul 26 2007
RH
RA 192.20.225.32
RU
RI





From petsie at dordos.net  Fri Jul 27 03:38:21 2007
From: petsie at dordos.net (Peter Pein)
Date: Fri, 27 Jul 2007 03:38:21 +0200
Subject: a propos divisors...
In-Reply-To: <46A812CD.60303@dordos.net>
References: <46A50268.3040307@dordos.net> <46A812CD.60303@dordos.net>
Message-ID: <46A94C8D.4090109@dordos.net>

Peter Pein schrieb:
> 
> Although the interst seems to be bounded by a fairly small n (0 until
> now), I would like to ask you if you could please help me.
>
....

> 
> Thank you for your attention and in advance for CPU-time,
> 
> Peter
> 

Well, there are other groups on the net...

THX for ignoring me completely
Peter



Peter,   I at least did not ignore you!   I saved
all your messages to be read later. 

(I just didn't reply yet!)

Neil



>%I A000001
>%S A000001 3, 5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22,
>23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42,
>43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63
>%N A000001 Those integers for which a smaller positive integer exists
>which has the same number of divisors
>%F A000001 a(n)= n-th element of {x&gt;0, there exists a k with
>0&lt;k&lt;x and the same number of divisors as x)
>%e A000001 a(4)=8 because it is the fourth integer for which a smaller
>integer with the same number of divisors exists (after 3, 5 and 7).
>divisors of 8 are 1,2,4,8 which are four and the divisors of 6 which is
>less than 8 are (1, 2, 3, 6) which are four.
>%t A000001 Clear[tmp];
>Function[n, If[Head[#1] === tmp, #1 = n; Unevaluated[Sequence[]], n] &
>   [tmp[DivisorSigma[0, n]]]] /@ Range[64]
>%Y A000001 Cf. A069822, A131902-A131908
>%O A000001 1
>%K A000001 ,easy,nonn,
>%A A000001 Peter Pein (petsie at dordos.net), Jul 26 2007
>RH
>RA 192.20.225.32
>RU
>RI

Appears to be the complement of A007416.

Tony




From njas at research.att.com  Fri Jul 27 04:02:35 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Thu, 26 Jul 2007 22:02:35 -0400 (EDT)
Subject: a propos divisors...
In-Reply-To: <46A93BB9.1010209@dordos.net>
References: <46A50268.3040307@dordos.net> <46A812CD.60303@dordos.net>
Message-ID: <200707270202.l6R22ZH92491631@fry.research.att.com>

On 7/25/07, Peter Pein <petsie at dordos.net> wrote:

> a) n such there is at least one x<n such that sigma(r,x)=sigma(r,n),
> because there exists such a series for r=1 (A069822)
>
> and
>
> b) a(n)=min({k>0: sigma(r,k)=sigma(r,n)})

[...]

> but for r = 3 the air becomes very thin. If I did not make a mistake,
> this seq starts(!):
>
> 194315, 295301, 590602, 1181204, 1476505, 1886920, 2067107, 2362408,
> 2526095, 2953010, 3248311, 3691985, 3838913, 4134214, 4469245, 4724816,
> 5020117, 5610719, 5635135, 5906020

[...]

> And if you want to make me really happy (I have had birthday on July
> 24th ;-) (http://www.stevesbeatles.com/songs/when_im_sixty_four.asp this
> age will be reached in 20 years, but the symptoms... )):

My congratulations!
Better late than never ;)

>  If you've got Mathematica and more RAM (4GB or so) than I do (1.5 GB),
> could you please run this code  with, say nmax=10 or 20 million? On my
> machine it swapped heavily with nmax=6 million and I had to kill
> MathKernel as I tried nmax=10^7. The lines above took ~181 seconds to
> evaluate (nmax=10^7 has been stopped by me after 15 minutes). I do not
> expect any runtimes of more than 7 minutes. Would this be possible, please?

These are the values below 10^7 that I got with my C++ program using
LiDIA library:

194315 184926
295301 291741
590602 583482
1181204 1166964
1476505 1458705
1886920 1880574
2067107 2042187
2362408 2333928
2526095 2404038
2953010 2917410
3248311 3209151
3691985 3513594
3838913 3792633
4134214 4084374
4469245 4253298
4724816 4667856
5020117 4959597
5610719 5543079
5635135 5362854
5906020 5834820
6023765 5732706
6496622 6418302
6791923 6710043
7382525 7293525
7677826 7585266
7966915 7581966
8268428 8168748
8355545 7951818
8563729 8460489
9132805 8691522
9449632 9335712

Each here line contains a pair:
n k
such that sigma(3,k)=sigma(3,n) and k=a(n) for r=3 (following your
notations above).

I will let my program to run for a couple more days to reach 10^8 bound.

> (AFAIK there exists no kind of "inverse
> function" to sigma(r,n) w.r.t. n which could be calculated without this
> brute-force method).

I disagree with this statement. There is a more or less clever way to
reconstruct the inverse of m=sigma(r,n) w.r.t. n, using integer
factorization of m and a kind of brute-force but of the magnitude of
the number of divisors of m.

Regards,
Max




From petsie at dordos.net  Fri Jul 27 16:32:12 2007
From: petsie at dordos.net (Peter Pein)
Date: Fri, 27 Jul 2007 16:32:12 +0200
Subject: [Fwd: SEQ FROM Peter Pein]
In-Reply-To: <p0624083ec2cf2f4f87a8@[192.168.1.43]>
References: <46A93BB9.1010209@dordos.net> <p0624083ec2cf2f4f87a8@[192.168.1.43]>
Message-ID: <46AA01EC.2070906@dordos.net>

T. D. Noe schrieb:
>> %I A000001
>> %S A000001 3, 5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22,
>> 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42,
>> 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63
>> %N A000001 Those integers for which a smaller positive integer exists
>> which has the same number of divisors
>> %F A000001 a(n)= n-th element of {x&gt;0, there exists a k with
>> 0&lt;k&lt;x and the same number of divisors as x)
>> %e A000001 a(4)=8 because it is the fourth integer for which a smaller
>> integer with the same number of divisors exists (after 3, 5 and 7).
>> divisors of 8 are 1,2,4,8 which are four and the divisors of 6 which is
>> less than 8 are (1, 2, 3, 6) which are four.
>> %t A000001 Clear[tmp];
>> Function[n, If[Head[#1] === tmp, #1 = n; Unevaluated[Sequence[]], n] &
>>   [tmp[DivisorSigma[0, n]]]] /@ Range[64]
>> %Y A000001 Cf. A069822, A131902-A131908
>> %O A000001 1
>> %K A000001 ,easy,nonn,
>> %A A000001 Peter Pein (petsie at dordos.net), Jul 26 2007
>> RH
>> RA 192.20.225.32
>> RU
>> RI
> 
> Appears to be the complement of A007416.
> 
> Tony
> 
Thank you, Tony for this hint.
I ask a bit too late, but:
is it common to let the superseeker have a look at the sequence before
publishing it? I have published seven sequences last night. At the rate
of one request per hour this would last too long.

Peter



* Peter Pein <petsie at dordos.net> [Jul 27. 2007 17:36]:
> [...]

> Thank you, Tony for this hint.
> I ask a bit too late, but:
> is it common to let the superseeker have a look at the sequence before
> publishing it? I have published seven sequences last night. At the rate
> of one request per hour this would last too long.
> 
> Peter

I strongly support to super-seek all new sequences before having them
in the database.

How much resources does one super-seek take?  (CPU & memory)



Peter,  I have put you on the list of "good guys"
who are not subject to the one per hour limit!
Neil




From arndt at jjj.de  Fri Jul 27 17:40:29 2007
From: arndt at jjj.de (Joerg Arndt)
Date: Fri, 27 Jul 2007 17:40:29 +0200
Subject: [Fwd: SEQ FROM Peter Pein]
In-Reply-To: <46AA01EC.2070906@dordos.net>
References: <46A93BB9.1010209@dordos.net> <p0624083ec2cf2f4f87a8@[192.168.1.43]> <46AA01EC.2070906@dordos.net>
Message-ID: <20070727154029.GA22447@amd32.purzl.net>

While preparing to submit to OEIS the decimal digits of two constant
used in the Ramanujan-Lodge Harmonic Number approximation and the
DeTemple-Wang Harmonic Number approximation, I ran into trouble.

I must be doing something really stupid.

Both the low-res Google calculator and the high-res WIMS calculator tell me that
	
((12 * gamma) - 11 - (12 * ln(2))) / (1 - gamma - ln(2^0.5)) = -162.5909

where gamma is Euler's constant.

But Theorem 6 of the arXiv citation, formula (1.13), page 6, insists
that this constant is roughly 1.12150934.

http://arxiv.org/pdf/0707.3950
    Title: Ramanujan's Harmonic Number Expansion into NegativePowers
of a Triangular Number
    Authors: Mark B. Villarino
    Comments: sharp error estimates and general formulas for
Ramanujan's harmonic number expansion
    Subjects: Classical Analysis and ODEs (math.CA); General
Mathematics (math.GM)

    An algebraic transformation of the DeTemple-Wang half-integer
approximation to the harmonic series produces the general formula and
error estimate for the Ramanujan expansion for the nth harmonic number
into negative powers of the nth triangular number. We also discuss the
history of the Ramanujan expansion for the nth harmonic number as well
as sharp estimates of its accuracy, with complete proofs, and we
compare it with other approximative formulas.

As Harminic numbers and Ramanujan appear to popular on EIS, may I ask
for someone to help me in my befuddlement?




From jvospost3 at gmail.com  Fri Jul 27 20:48:04 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Fri, 27 Jul 2007 11:48:04 -0700
Subject: Ramanujan-Lodge's Harmonic Number approximation
In-Reply-To: <563222.67756.qm@web86610.mail.ukl.yahoo.com>
References: <5542af940707270852l3682b86dxdcd8d6dbdabd7ee4@mail.gmail.com>
	 <563222.67756.qm@web86610.mail.ukl.yahoo.com>
Message-ID: <5542af940707271148j2dfc833al93434b370451e97d@mail.gmail.com>

Thank you, Ray and Martin.

I have emailed Mark B. Villarino, using the most recent Costa Rica
email address I could find by Googling (oddly, one is not given in the
arXiv paper), saying, before including your comments (but not giving
out your email addresses nor that of Seqfans):

I enjoyed your paper (and several earlier papers as well on Decartes'
Perfect Lens,  On the Archimedean or Semiregular Polyhedra,  Mertens'
Proof of Mertens' Theorem, ...), and initiated an online conversation
that provides you a correction and a simplification.




From martin_n_fuller at btinternet.com  Fri Jul 27 19:49:50 2007
From: martin_n_fuller at btinternet.com (Martin Fuller)
Date: Fri, 27 Jul 2007 18:49:50 +0100 (BST)
Subject: Ramanujan-Lodge's Harmonic Number approximation
In-Reply-To: <5542af940707270852l3682b86dxdcd8d6dbdabd7ee4@mail.gmail.com>
Message-ID: <563222.67756.qm@web86610.mail.ukl.yahoo.com>

It looks like a typo in the paper.  It should be:
((12 * gamma) - 11 + (12 * ln(2^0.5))) / (1 - gamma - ln(2^0.5))

or more simply:
1 / (1 - gamma - ln(2^0.5)) - 12

The second constant is more simply written as:
1 / (1 - gamma - ln(3/2)) - 54

--- Jonathan Post <jvospost3 at gmail.com> wrote:

> While preparing to submit to OEIS the decimal digits of two constant
> used in the Ramanujan-Lodge Harmonic Number approximation and the
> DeTemple-Wang Harmonic Number approximation, I ran into trouble.
> 
> I must be doing something really stupid.
> 
> Both the low-res Google calculator and the high-res WIMS calculator
> tell me that
> 	
> ((12 * gamma) - 11 - (12 * ln(2))) / (1 - gamma - ln(2^0.5)) =
> -162.5909
> 
> where gamma is Euler's constant.
> 
> But Theorem 6 of the arXiv citation, formula (1.13), page 6, insists
> that this constant is roughly 1.12150934.
> 
> http://arxiv.org/pdf/0707.3950
>     Title: Ramanujan's Harmonic Number Expansion into NegativePowers
> of a Triangular Number
>     Authors: Mark B. Villarino
>     Comments: sharp error estimates and general formulas for
> Ramanujan's harmonic number expansion
>     Subjects: Classical Analysis and ODEs (math.CA); General
> Mathematics (math.GM)
> 
>     An algebraic transformation of the DeTemple-Wang half-integer
> approximation to the harmonic series produces the general formula and
> error estimate for the Ramanujan expansion for the nth harmonic
> number
> into negative powers of the nth triangular number. We also discuss
> the
> history of the Ramanujan expansion for the nth harmonic number as
> well
> as sharp estimates of its accuracy, with complete proofs, and we
> compare it with other approximative formulas.
> 
> As Harminic numbers and Ramanujan appear to popular on EIS, may I ask
> for someone to help me in my befuddlement?
> 




I'm curious how a sieve, similar to the sieve of Erosthanese, would
"perform".  The basic idea is this: suppose that n is some number that you
want to factor. Assume that you can map, for x = 0 to some upper bound less
than the smallest factor of n, n == a mod x --> f == b mod x, where f is a
factor of n. Say, for instance:

n == 2 mod 3, therefore f == 1 mod 3
n == 1 mod 4, therefore f == 3 mod 4
n == 4 mod 5, therefore f == 2 mod 5

& etc.

Thus we could establish a "profile" for f, which can be turned into a sieve
by finding all integers congruent to 1 mod 3, then a subset congruent to 3
mod 4, then a subset of that congruent to 2 mod 5, etc. My question is how
"quickly" can we narrow in on possible values for f by using this sieve? I
think something like this may be possible for values of n with certain
properties, but I'm not sure how well it'd perform.







From aplewe at sbcglobal.net  Fri Jul 27 21:28:30 2007
From: aplewe at sbcglobal.net (Andrew Plewe)
Date: Fri, 27 Jul 2007 12:28:30 -0700
Subject: A theoretical question -- sieving via n mod x
In-Reply-To: <5542af940707271148j2dfc833al93434b370451e97d@mail.gmail.com>
References: <5542af940707270852l3682b86dxdcd8d6dbdabd7ee4@mail.gmail.com>
	 <563222.67756.qm@web86610.mail.ukl.yahoo.com>
	 <5542af940707271148j2dfc833al93434b370451e97d@mail.gmail.com>
Message-ID: <200707271935.l6RJZCci090307@shiva.jussieu.fr>

I just submitted 4 related sequences, A131915 through A131918, which
are the digits (1st 100) of the decimal expansions of, and the
continued fraction expansions (100 integers) of the two formulae, in
all cases crediting Martin Fuller's simplifications and/or corrections
of Mark B. Villarino's paper.




From jvospost3 at gmail.com  Sat Jul 28 02:38:51 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Fri, 27 Jul 2007 17:38:51 -0700
Subject: Ramanujan-Lodge's Harmonic Number approximation
In-Reply-To: <5542af940707271735t4ea8f890y39802dd9c657193@mail.gmail.com>
References: <5542af940707270852l3682b86dxdcd8d6dbdabd7ee4@mail.gmail.com>
	 <563222.67756.qm@web86610.mail.ukl.yahoo.com>
	 <5542af940707271148j2dfc833al93434b370451e97d@mail.gmail.com>
	 <5542af940707271735t4ea8f890y39802dd9c657193@mail.gmail.com>
Message-ID: <5542af940707271738m4533a2e7rbe837493872723c8@mail.gmail.com>

Synchronicity strikes!  At the same minute, I received this email from
the author of the arXiv paper which Ray Chandler helped me determine
had a typo, and Martin Fuller corrected and simplified the formulae:

==============

Dear Jonathan Vos Post,

Thank you so very much for your letter.  You are quite right, and I
have incorporated the correction/simplification into my preprint.  It
should appear on lanl.xxx on Monday.

I am very grateful for the trouble you took and, as a naturally
prejudiced author, I thank you for your kind words about my other preprints.

The paper on semiregular polyhedra will appear in Elemente der
Mathematik...it has been accepted for publication as part of the
"Euler year."

The paper on "The Probability of a Run" was published in the March,
2007,  issue of the Mathematical Gazette.

Finally, the paper on the accuracy or Ramanujan's approximation to the
arc length of an ellipse was published in the online journal JIPAM
(Journal of Inequalities in Pure and Applied Mathematics) in January
of 2006.

I would be delighted to join into any further dialogue about the paper.

Again, thank you very much.

Sincerely yours,
MARK B. VILLARINO

MARK B. VILLARINO
Escuela de matem?tica
Universidad de Costa Rica
San Jos?, Costa RicaTelephone: 506-857-4679
email: mvillari at cariari.ucr.ac.cr
email: mark.villarino at gmail.com

==============





From franktaw at netscape.net  Sat Jul 28 02:43:56 2007
From: franktaw at netscape.net (franktaw at netscape.net)
Date: Fri, 27 Jul 2007 20:43:56 -0400
Subject: A theoretical question -- sieving via n mod x
In-Reply-To: <200707271935.l6RJZCci090307@shiva.jussieu.fr>
References: <200707271935.l6RJZCci090307@shiva.jussieu.fr>
Message-ID: <8C99EC783AE068B-5F0-7232@mblk-d13.sysops.aol.com>

Assuming that evaluating your function is fast, this should be very 
efficient.  Combining results in different moduli is essentially just 
Euclid's algorithm (for the GCD) -- which is quite fast -- followed by 
a couple of multiplications and an addition.

Franklin T. Adams-Watters

-----Original Message-----
From: Andrew Plewe <aplewe at sbcglobal.net>

I'm curious how a sieve, similar to the sieve of Erosthanese, would
"perform".  The basic idea is this: suppose that n is some number that 
you
want to factor. Assume that you can map, for x = 0 to some upper bound 
less
than the smallest factor of n, n == a mod x --> f == b mod x, where f 
is a
factor of n. Say, for instance:

n == 2 mod 3, therefore f == 1 mod 3
n == 1 mod 4, therefore f == 3 mod 4
n == 4 mod 5, therefore f == 2 mod 5

& etc.

Thus we could establish a "profile" for f, which can be turned into a 
sieve
by finding all integers congruent to 1 mod 3, then a subset congruent 
to 3
mod 4, then a subset of that congruent to 2 mod 5, etc. My question is 
how
"quickly" can we narrow in on possible values for f by using this 
sieve? I
think something like this may be possible for values of n with certain
properties, but I'm not sure how well it'd perform.

________________________________________________________________________
Check Out the new free AIM(R) Mail -- Unlimited storage and 
industry-leading spam and email virus protection.



There are three seqs titled "Number of weighted voting procedures"
http://www.research.att.com/~njas/sequences/A005254
http://www.research.att.com/~njas/sequences/A005256
http://www.research.att.com/~njas/sequences/A005257

Could somebody define the term "weighted voting procedure"
and add the information to differentiate the seqs?




Hello seqfans,

Here I see two entries with the same name:

A030225 Number of n-celled polyhexes (hexagonal polyominoes) with bilateral symmetry.
A002215 Number of polyhexes with n hexagons, having reflectional symmetry. 


Probably, the second one should be about "restricted" polyhexes, BUT, there are two entries about restricted polyhexes:

A002216 Harary-Read numbers: restricted hexagonal polyominoes (cata-polyhexes) with n cells. 
A002212 Number of restricted hexagonal polyominoes with n cells. 


On top of that, if a2216 is the number of cata-polyhexes, then what is
A038142 Number of planar cata-polyhexes with n cells.

I am completely confused.

Best, Tanya


_________________________________________________________________
Need personalized email and website? Look no further. It's easy
with Doteasy $0 Web Hosting! Learn more at www.doteasy.com



You did such a great job removing duplicate sequences. How about duplicate definitions:

A002212 Number of restricted hexagonal polyominoes with n cells.
A005963 Number of restricted hexagonal polyominoes with n cells. 

Tanya


_________________________________________________________________
Need personalized email and website? Look no further. It's easy
with Doteasy $0 Web Hosting! Learn more at www.doteasy.com



A001168 Number of fixed polyominoes with n cells.
A006762 Number of fixed polyominoes with n cells. 


_________________________________________________________________
Need personalized email and website? Look no further. It's easy
with Doteasy $0 Web Hosting! Learn more at www.doteasy.com




Let me remind folks that the primary purpose
of the OEIS is to give pointers to the literature,
you can find out who else has studied the same sequence.

I don't claim to give precise definitions for every single

If you find two sequences with the same definition
("Related to the enumeration of ***", say),
feel free to track down the references and

Neil




From arndt at jjj.de  Sat Jul 28 12:06:39 2007
From: arndt at jjj.de (Joerg Arndt)
Date: Sat, 28 Jul 2007 12:06:39 +0200
Subject: weighted voting sequences
Message-ID: <20070728100639.GA15382@amd32.purzl.net>

so that when you come across a sequence in your work, 
sequence - that would take more time than I have available.
send me more precise definitions.
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Message-ID: <5542af940707281505n357500fbu2fada9399c6b518c at mail.gmail.com>
Date: Sat, 28 Jul 2007 15:05:59 -0700
From: "Jonathan Post" <jvospost3 at gmail.com>
To: "Max Alekseyev" <maxale at gmail.com>
Subject: Re: definition of anti-divisor
Cc: "Maximilian Hasler" <maximilian.hasler at gmail.com>,
   "Sequence Fans" <seqfan at ext.jussieu.fr>, jvospost2 at yahoo.com
In-Reply-To: <5542af940707242126q68201c55pf5c674ce6848194f at mail.gmail.com>
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References: <200707211141.l6LBf9dn795643 at fry.research.att.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b at mail.gmail.com>
	 <5542af940707221117t4e411e40o840c9228f2e9b7f8 at mail.gmail.com>
	 <3c3af2330707231221y5690ec3cm473f38b5f2bc6adc at mail.gmail.com>
	 <5542af940707231709n330ca0ffkc581bb34984b1154 at mail.gmail.com>
	 <5542af940707231811t4044fa14rcdc33c9a227a9562 at mail.gmail.com>
	 <d3dac270707231857y6dae1f1cq27367b56568301cb at mail.gmail.com>
	 <d3dac270707231929r558496f6t36dfb40745bb02fe at mail.gmail.com>
	 <d3dac270707232202q7d6d4198t259627c7b9977368 at mail.gmail.com>
	 <5542af940707242126q68201c55pf5c674ce6848194f at mail.gmail.com>
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NEW SEQUENCE FROM Jonathan Vos Post

%I A000001
%S A000001 3, 18, 1728, 679477248
%N A000001 Anti-primorials, partial products of anti-primes A092680.
%C A000001 This is to primorial (A002110) as anti-prime (A092680) is
to prime (A000040). Max Alekseyev <maxale at gmail.com> points out
 that every element of A066466, except 4, must be of the form 3*2^k
such that 3*2^(k+1)-1, 3*2^(k+1)+1 are twin primes. There no such new
k+1 (i.e., except known 1,2,6,18) below 1000.
In other words, 3*2^n - 1, 3*2^n + 1 are twin primes for n=1,2,6,18.
 According to these tables:
http://www.prothsearch.net/riesel.html
http://www.prothsearch.net/riesel2.html
there are no other such n up to 1200000.
Therefore, the next element of A066466 (if it exists) is greater than
3*2^1200000 ~= 10^361236. Hence the next element of the
anti-primorials (if it exists) is greater than 679477248 *3*2^1200000
~= 679477248 *
 10^361236 ~= 6 * 10^361245.
%D A000001 1.
%F A000001 a(n) = PRODUCT[k = 1..n] A092680(k).
%e A000001 a(1) = 3.
a(2) = 3 * 6 = 18.
a(3) = 3 * 6 * 96 = 1728.
a(4) = 3 * 6 * 96 * 393216 = 679477248.
%Y A000001 Cf. A000040, A002110, A092680, A130874.
%O A000001 1,1
%K A000001 ,nonn,unkn,
%A A000001 Jonathan Vos Post (jvospost2 at yahoo.com), Jul 28 2007
RH
RA 192.20.225.32




From petsie at dordos.net  Sun Jul 29 01:04:43 2007
From: petsie at dordos.net (Peter Pein)
Date: Sun, 29 Jul 2007 01:04:43 +0200
Subject: a propos divisors...
In-Reply-To: <d3dac270707270520r26d2f151rf531267baad94f02@mail.gmail.com>
References: <46A50268.3040307@dordos.net> <46A812CD.60303@dordos.net> <d3dac270707270520r26d2f151rf531267baad94f02@mail.gmail.com>
Message-ID: <46ABCB8B.2090809@dordos.net>

Max Alekseyev schrieb:
> On 7/25/07, Peter Pein <petsie at dordos.net> wrote:
> 
>> a) n such there is at least one x<n such that sigma(r,x)=sigma(r,n),
>> because there exists such a series for r=1 (A069822)
>>
>> and
>>
>> b) a(n)=min({k>0: sigma(r,k)=sigma(r,n)})
> 
> [...]
> 
>> but for r = 3 the air becomes very thin. If I did not make a mistake,
>> this seq starts(!):
>>
>> 194315, 295301, 590602, 1181204, 1476505, 1886920, 2067107, 2362408,
>> 2526095, 2953010, 3248311, 3691985, 3838913, 4134214, 4469245, 4724816,
>> 5020117, 5610719, 5635135, 5906020
> 
> [...]
> 
>> And if you want to make me really happy (I have had birthday on July
>> 24th ;-) (http://www.stevesbeatles.com/songs/when_im_sixty_four.asp this
>> age will be reached in 20 years, but the symptoms... )):
> 
> My congratulations!
> Better late than never ;)

Thank you!

> 
>>  If you've got Mathematica and more RAM (4GB or so) than I do (1.5 GB),
>> could you please run this code  with, say nmax=10 or 20 million? On my
>> machine it swapped heavily with nmax=6 million and I had to kill
>> MathKernel as I tried nmax=10^7. The lines above took ~181 seconds to
>> evaluate (nmax=10^7 has been stopped by me after 15 minutes). I do not
>> expect any runtimes of more than 7 minutes. Would this be possible, please?
> 
> These are the values below 10^7 that I got with my C++ program using
> LiDIA library:
> 
> 194315 184926
> 295301 291741
> 590602 583482
> 1181204 1166964
> 1476505 1458705
> 1886920 1880574
> 2067107 2042187
> 2362408 2333928
> 2526095 2404038
> 2953010 2917410
> 3248311 3209151
> 3691985 3513594
> 3838913 3792633
> 4134214 4084374
> 4469245 4253298
> 4724816 4667856
> 5020117 4959597
> 5610719 5543079
> 5635135 5362854
> 5906020 5834820
> 6023765 5732706
> 6496622 6418302
> 6791923 6710043
> 7382525 7293525
> 7677826 7585266
> 7966915 7581966
> 8268428 8168748
> 8355545 7951818
> 8563729 8460489
> 9132805 8691522
> 9449632 9335712
> 
> Each here line contains a pair:
> n k
> such that sigma(3,k)=sigma(3,n) and k=a(n) for r=3 (following your
> notations above).
> 
> I will let my program to run for a couple more days to reach 10^8 bound.
> 
>> (AFAIK there exists no kind of "inverse
>> function" to sigma(r,n) w.r.t. n which could be calculated without this
>> brute-force method).
> 
> I disagree with this statement. There is a more or less clever way to
> reconstruct the inverse of m=sigma(r,n) w.r.t. n, using integer
> factorization of m and a kind of brute-force but of the magnitude of
> the number of divisors of m.
> 
> Regards,
> Max
> 
Dear Max,

:-) thank you very much! :-)

I just entered the extension to A13190{7|8} and of course mentioned your
name. When your program gives more results, please extend the sequences
or if you've got a lot of numbers, send Neil a b-file.

The method you mention sounds mangeable - I will look for that algorithm.

Thanks again,
Peter



Dear seqfans,
I'd appreciate information (including references) 
about conditions (necessary and/or sufficient) on 
a sequence of positive numbers a1, a2, a3, ... 
in order that their Hankel matrix be positive 
definite. 
Thanks,
Emeric



Dear seqfans,
I'd appreciate information (including references) 
about conditions (necessary and/or sufficient) on 
a sequence of positive numbers a1, a2, a3, ... 
in order that their Hankel matrix be positive 
definite. 
Thanks,
Emeric




From deutsch at duke.poly.edu  Sun Jul 29 06:12:09 2007
From: deutsch at duke.poly.edu (deutsch)
Date: Sun, 29 Jul 2007 04:12:09 GMT
Subject: positive definiteness of Hankel matrices
In-Reply-To: <200707280825.AA3347841186@TanyaKhovanova.com>
References: <200707280825.AA3347841186@TanyaKhovanova.com>
Message-ID: <46ac1399.275.718.7202@duke.poly.edu>

A006762 is counting only polyominoes that are strictly 2-dimensional - 
it excludes those where all the squares are in a single line.  Thus, 
for n>1, A006762(n) = A001168(n) - 2.

Franklin T. Adams-Watters

-----Original Message-----
From: Tanya Khovanova <tanyakh at TanyaKhovanova.com>

A001168 Number of fixed polyominoes with n cells.
A006762 Number of fixed polyominoes with n cells.

________________________________________________________________________
Check Out the new free AIM(R) Mail -- Unlimited storage and 
industry-leading spam and email virus protection.




From neoneye at gmail.com  Sun Jul 29 19:13:08 2007
From: neoneye at gmail.com (Simon Strandgaard)
Date: Sun, 29 Jul 2007 19:13:08 +0200
Subject: Sequence A112088 Motivation and Example?
In-Reply-To: <46A90302.8020905@web.de>
References: <4489D93F.3080902@web.de>
	 <020401c68c0a$9470a650$6900000a@mcraeclan.com>
	 <02a801c68c1c$1421f320$6900000a@mcraeclan.com>
	 <d3dac270606092351x3c02aeeo84c20b6871eb7e68@mail.gmail.com>
	 <d3dac270707261223l69633b85n2277d5adfc2f5869@mail.gmail.com>
	 <46A90302.8020905@web.de>
Message-ID: <df1390cc0707291013q696575d9hd69b17382edb8141@mail.gmail.com>

On 7/26/07, Rainer Rosenthal <r.rosenthal at web.de> wrote:
> Max Alekseyev wrote:
> > On 6/9/06, Max <maxale at gmail.com> wrote:
> >>On 6/9/06, Graeme McRae <g_m at mcraefamily.com> wrote:
> >>>It took me quite some time to understand how Ranier Rosenthal is counting
>                                                 Rainer (please)
> >>>rounds, before I obtained A112088 as the number of rounds necessary to kill
> >>>all but one of n players of the Josephus Game with every third man out.
>
> Many many thanks for this post!
>
> Some days ago I found my last scribblings, which were meant for
> another comment to A112088. But it's all that long ago ... I
> didn't understand anything of what I wrote :-(
>
> I'm going to print your mail now and maybe I will find the
> energy to get back to all these lovely round-countings.
>
> As far as I remember I wasn't able to really grasp the idea of
> the original submitter (Simon Strandgaard). From the discussions
> in de.sci.mathematik and de.rec.denksport I remember that there
> were some remarks giving hints to Knuth. Oops ... long time ago.

I wrote a step-by-step guide to how a112088 works, see
http://www.research.att.com/~njas/sequences/a112088.html

maybe it can be useful to you?

-- 
Simon Strandgaard




From Eric.Angelini at kntv.be  Mon Jul 30 00:11:35 2007
From: Eric.Angelini at kntv.be (Eric Angelini)
Date: Mon, 30 Jul 2007 00:11:35 +0200
Subject: Substrings in A046043 
Message-ID: <F05775148D000C4B9321A5113D53CB7D50D245@KNTV-SERVER01.kntv.local>

Hello SeqFans,

Integers of A046043 are interpreted like this:

0123
1210 = one 0 two 1 one 2 zero 3 -> one,two,one,zero = 1210

0123
2020 = two 0 zero 1 two 2 zero 3 -> two,zero,two,zero = 2020

01234
21200 = two 0 one 1 two 2 zero 3 zero 4 -> two,one,two,zero,zero = 21200

0123456
3211000 = three 0 two 1 one 2 one 3 zero 4 zero 5 zero 6 -> etc. = 3211000 

...

0123456789
6210001000 = six 0 two 1 one 2 zero 3 zero 4 zero 5 one 6 zero 7 zero 8 zero 9

Now if we add an extra string "10" in the upper line we could perhaps go on in
the sequence like this:

012345678910
53110100002

... meaning that we have in the integer 53110100002:

- five (sub)strings "0"
- three strings "1"
- one string  2
- one string  3
- zero string 4
- one string  5
- zero string 6
- zero string 7
- zero string 8
- zero string 9
- two strings 10

What would be the next integers in the sequence?

Best,
E.

 


 

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From Eric.Angelini at kntv.be  Mon Jul 30 00:45:48 2007
From: Eric.Angelini at kntv.be (Eric Angelini)
Date: Mon, 30 Jul 2007 00:45:48 +0200
Subject: =?iso-8859-1?Q?RE=A0=3A_Substrings_in_A046043_?=
Message-ID: <F05775148D000C4B9321A5113D53CB7D50D246@KNTV-SERVER01.kntv.local>

...

> meaning that we have in the integer 53110100002 (...)

... The length of such an integer indicates obviously how many
(sub)strings have to be discribed:
53110100002 has 11 digits thus the 11 substrings "0", "1"... to "10"
must be described

> What would be the next integers in the sequence?

... I don't know if there is something before 62200010001:

012345678910
62200010001 

Best,
E.

 


 

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From qq-quet at mindspring.com  Mon Jul 30 16:23:05 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Mon, 30 Jul 07 08:23:05 -0600
Subject: Do any integers occur in both sequences?
In-Reply-To: <46ABCB8B.2090809@dordos.net>
References: <46A50268.3040307@dordos.net> <46A812CD.60303@dordos.net>
	 <d3dac270707270520r26d2f151rf531267baad94f02@mail.gmail.com>
	 <46ABCB8B.2090809@dordos.net>
Message-ID: <E1IFWBz-0000Pz-00@pop03.mail.atl.earthlink.net>

------=_Part_17373_17766144.1185831725052
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 7bit
Content-Disposition: inline

On 7/28/07, Peter Pein <petsie at dordos.net> wrote:

> > Each here line contains a pair:
> > n k
> > such that sigma(3,k)=sigma(3,n) and k=a(n) for r=3 (following your
> > notations above).
> >
> > I will let my program to run for a couple more days to reach 10^8 bound.

I stopped the program after it reached 7*10^7 as it started to eat too
much memory.
The results are attached. Please format them and submit to OEIS as you
feel appropriate.

P.S. Eventually I've defended Ph.D. in Computer Science ;)

Regards,
Max

------=_Part_17373_17766144.1185831725052
Content-Type: text/plain; name=sigma3.txt; charset=ANSI_X3.4-1968
Content-Transfer-Encoding: base64
X-Attachment-Id: f_f4rhifo0
Content-Disposition: attachment; filename="sigma3.txt"

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MAo1OTY1MDgwMiA1ODkzMTY4Mgo1OTY1NDcwNSA1Njc3MjI4Mgo1OTk0NjEw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------=_Part_17373_17766144.1185831725052--




From petsie at dordos.net  Tue Jul 31 04:32:57 2007
From: petsie at dordos.net (Peter Pein)
Date: Tue, 31 Jul 2007 04:32:57 +0200
Subject: Do any integers occur in both sequences?
In-Reply-To: <E1IFWBz-0000Pz-00@pop03.mail.atl.earthlink.net>
References: <E1IFWBz-0000Pz-00@pop03.mail.atl.earthlink.net>
Message-ID: <46AE9F59.9010203@dordos.net>

Leroy Quet schrieb:
> I have just submitted these two interdependent sequences (So don't look 
> for them in the database yet):
> 
>> %I A131937
>> %S A131937 1,4,8,14,21,29,38,49,61
>> %N A131937 a(1)=1; a(2)=4. a(n) = a(n-1) + (nth positive integer which 
>> does not occur in sequence A131938).
>> %e A131937 A131938: 2,5,10,16,23,32,42,53,...
>> Positive integers not in A131938: 1,3,4,6,7,8,9,11,...
>> So A131937(8) = A131937(7) + 11 = 49.
>> %Y A131937 A131938
>> %O A131937 1
>> %K A131937 ,more,nonn,
> 
>> %I A131938
>> %S A131938 2,5,10,16,23,32,42,53,65,78,93,109
>> %N A131938 a(1)=2; a(2)=5. a(n) = a(n-1) + (nth positive integer which 
>> does not occur in sequence A131937).
>> %e A131938 A131937: 1,4,8,14,21,29,...
>> Positive integers not in A131937: 2,3,5,6,7,9,10,11,...
>> So A131938(8) = A131938(7) + 11 = 53.
>> %Y A131938 A131937
>> %O A131938 1
>> %K A131938 ,more,nonn,
> 
> 
> I have not thought about this too hard; so for all I know, the proof is 
> quite easy.
> 
> Do any positive integers occur in both A131937 and A131938?
> 
> Thanks,
> Leroy Quet
> 

Hello Leroy,

let's just do the first step (n=3) to extend these lists:
(let A131937=:l1 and A131938=:l2)

I get
Complement[posint, l2]={1,3,4,6,7,8,9,...}
Therefore the third positive integer which is not in l2 is 4 again and
l1 starts {1, 4, 4, 6, 8, 10, 12}
and l2 begins {2, 5, 5, 7, 9, 11, 13}

The Mma-Code and its output:

each "grouped output" consists of l1 & l2 so far, n followed by the
complement of (the begin of) N w.r.t. l2 and l1 resp.; you can count the
elements using the latter.

In[26]:=
list1={1,4};list2={2,5};n=3;
Do[
    Print[{list1,list2}];
    Print[n," ",Complement[Range[15],#]&/@{list2,list1}];
    AppendTo[list1,Part[Complement[Range[n+Length[list2]],list2],n]];
    AppendTo[list2,Part[Complement[Range[n+Length[list1]],list1],n]];
    n++;
    ,{5}];
list1
list2
Intersection[%%,%]

 From In[26]:=
{{1,4},{2,5}}
 From In[26]:=
3 {{1,3,4,6,7,8,9,10,11,12,13,14,15},
   {2,3,5,6,7,8,9,10,11,12,13,14,15}}
 From In[26]:=
{{1,4,4},{2,5,5}}
 From In[26]:=
4 {{1,3,4,6,7,8,9,10,11,12,13,14,15},
   {2,3,5,6,7,8,9,10,11,12,13,14,15}}
 From In[26]:=
{{1,4,4,6},{2,5,5,7}}
 From In[26]:=
5 {{1,3,4,6,8,9,10,11,12,13,14,15},
   {2,3,5,7,8,9,10,11,12,13,14,15}}
 From In[26]:=
{{1,4,4,6,8},{2,5,5,7,9}}
 From In[26]:=
6 {{1,3,4,6,8,10,11,12,13,14,15},
   {2,3,5,7,9,10,11,12,13,14,15}}
 From In[26]:=
{{1,4,4,6,8,10},{2,5,5,7,9,11}}
 From In[26]:=
7 {{1,3,4,6,8,10,12,13,14,15},
   {2,3,5,7,9,11,12,13,14,15}}
Out[28]= (* A131937 *)
{1,4,4,6,8,10,12}
Out[29]= (* A131938 *)
{2,5,5,7,9,11,13}
Out[30]= (* intersection *)
{}

These are the first elements of l1 and l2 after 100 iterations:

{1,4,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,\
54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,\
104,106,108,110,112,114,116,118,120,122,124,126,128,130,132,134,136,138,140,\
142,144,146,148,150,152,154,156,158,160,162,164,166,168,170,172,174,176,178,\
180,182,184,186,188,190,192,194,196,198,200,202}

{2,5,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,\
55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,97,99,101,103,\
105,107,109,111,113,115,117,119,121,123,125,127,129,131,133,135,137,139,141,\
143,145,147,149,151,153,155,157,159,161,163,165,167,169,171,173,175,177,179,\
181,183,185,187,189,191,193,195,197,199,201,203}

and the intersection is empty.

Looks like the algorithm separates odd and even numbers.

Because the algorithm is very inefficient, I've chosen 10^4 as maximum
number of iterations:

list1={1,4};list2={2,5};n=3;
Do[
    AppendTo[list1,Part[Complement[Range[n+Length[list2]],list2],n]];
    AppendTo[list2,Part[Complement[Range[n+Length[list1]],list1],n]];
    n++;
    ,{10^4}];
Intersection[list1,list2]

--> {}

and the lists (almost) are the evens and the odds:

Take[#,-5]&/@{list1,list2}
{{19994,19996,19998,20000,20002},{19995,19997,19999,20001,20003}}

I guess, something went wrong in your definition or in my algorithm.


Best regards,
Peter




From petsie at dordos.net  Tue Jul 31 05:11:58 2007
From: petsie at dordos.net (Peter Pein)
Date: Tue, 31 Jul 2007 05:11:58 +0200
Subject: Do any integers occur in both sequences?
In-Reply-To: <E1IFWBz-0000Pz-00@pop03.mail.atl.earthlink.net>
References: <E1IFWBz-0000Pz-00@pop03.mail.atl.earthlink.net>
Message-ID: <46AEA87E.1010709@dordos.net>

Leroy Quet schrieb:
> I have just submitted these two interdependent sequences (So don't look 
> for them in the database yet):
> 
>> %I A131937
>> %S A131937 1,4,8,14,21,29,38,49,61
>> %N A131937 a(1)=1; a(2)=4. a(n) = a(n-1) + (nth positive integer which 
>> does not occur in sequence A131938).
>> %e A131937 A131938: 2,5,10,16,23,32,42,53,...
>> Positive integers not in A131938: 1,3,4,6,7,8,9,11,...
>> So A131937(8) = A131937(7) + 11 = 49.
>> %Y A131937 A131938
>> %O A131937 1
>> %K A131937 ,more,nonn,
> 
>> %I A131938
>> %S A131938 2,5,10,16,23,32,42,53,65,78,93,109
>> %N A131938 a(1)=2; a(2)=5. a(n) = a(n-1) + (nth positive integer which 
>> does not occur in sequence A131937).
>> %e A131938 A131937: 1,4,8,14,21,29,...
>> Positive integers not in A131937: 2,3,5,6,7,9,10,11,...
>> So A131938(8) = A131938(7) + 11 = 53.
>> %Y A131938 A131937
>> %O A131938 1
>> %K A131938 ,more,nonn,
> 
> 
> I have not thought about this too hard; so for all I know, the proof is 
> quite easy.
> 
> Do any positive integers occur in both A131937 and A131938?
> 
> Thanks,
> Leroy Quet
> 
Hi again,

even if you meant "n-th pos. int. which does yet not occur in either l1
or l2" (or does one say "..does yet neither occur in l1 nor in l2"?) I
get another result

list1={1,4};list2={2,5};n=3;
Do[
    Print[{list1,list2}];
    Print["n= ",n," ",Complement[Range[25],list2]];
    AppendTo[list1,Part[Complement[Range[3n ],Union[list1,list2]],n]];
    Print["n= ",n," ",Complement[Range[25],list1]];
    AppendTo[list2,Part[Complement[Range[3n],Union[list1,list2]],n]];
    n++;
    ,{5}];
list1
list2
Intersection[%%,%]


{{1,4},{2,5}}
n= 3 {1,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
n= 3 {2,3,5,6,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
{{1,4,7},{2,5,8}}
n= 4 {1,3,4,6,7,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
n= 4 {2,3,5,6,8,9,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
{{1,4,7,10},{2,5,8,11}}
n= 5 {1,3,4,6,7,9,10,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
n= 5 {2,3,5,6,8,9,11,12,14,15,16,17,18,19,20,21,22,23,24,25}
{{1,4,7,10,13},{2,5,8,11,14}}
n= 6 {1,3,4,6,7,9,10,12,13,15,16,17,18,19,20,21,22,23,24,25}
n= 6 {2,3,5,6,8,9,11,12,14,15,17,18,19,20,21,22,23,24,25}
{{1,4,7,10,13,16},{2,5,8,11,14,17}}
n= 7 {1,3,4,6,7,9,10,12,13,15,16,18,19,20,21,22,23,24,25}
n= 7 {2,3,5,6,8,9,11,12,14,15,17,18,20,21,22,23,24,25}

l1: {1,4,7,10,13,16,19}
l2: {2,5,8,11,14,17,20}
intersection: {}

And l1, l2 and intersection for the first 100 iterations are (sorry for
weird linewidths):

{1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52,
55, 58, 61, 64, 67, 70,
  73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118,
121, 124, 127, 130,
  133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172,
175, 178, 181, 184,
  187, 190, 193, 196, 199, 202, 205, 208, 211, 214, 217, 220, 223, 226,
229, 232, 235, 238,
  241, 244, 247, 250, 253, 256, 259, 262, 265, 268, 271, 274, 277, 280,
283, 286, 289, 292,
  295, 298, 301, 304}

{2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53,
56, 59, 62, 65, 68, 71,
  74, 77, 80, 83, 86, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119,
122, 125, 128, 131,
  134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173,
176, 179, 182, 185,
  188, 191, 194, 197, 200, 203, 206, 209, 212, 215, 218, 221, 224, 227,
230, 233, 236, 239,
  242, 245, 248, 251, 254, 257, 260, 263, 266, 269, 272, 275, 278, 281,
284, 287, 290, 293,
  296, 299, 302, 305}

{}

and after 10^4 iterations (without intersections), lists end :

In[121]:=
Take[#1, -5]& /@ {list1, list2}

Out[121]=
{{29992, 29995, 29998, 30001, 30004},
 {29993, 29996, 29999, 30002, 30005}}

This time we've got {x: x == i mod 3} with i = 1,2.

Testing the 10^4 iterations:

Union[Mod[#,3]]&/@{list1,list2}
--> {{1},{2}}

Could you please explain in detail, how you've got your sequences?

Peter




From maxale at gmail.com  Tue Jul 31 05:47:31 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Mon, 30 Jul 2007 20:47:31 -0700
Subject: Do any integers occur in both sequences?
In-Reply-To: <46AEA87E.1010709@dordos.net>
References: <E1IFWBz-0000Pz-00@pop03.mail.atl.earthlink.net>
	 <46AEA87E.1010709@dordos.net>
Message-ID: <d3dac270707302047i346165cred3b11e98db90bcd@mail.gmail.com>

On 7/30/07, Peter Pein <petsie at dordos.net> wrote:
> Leroy Quet schrieb:
> > I have just submitted these two interdependent sequences (So don't look
> > for them in the database yet):
> >
> >> %I A131937
> >> %S A131937 1,4,8,14,21,29,38,49,61
> >> %N A131937 a(1)=1; a(2)=4. a(n) = a(n-1) + (nth positive integer which
> >> does not occur in sequence A131938).
> >> %e A131937 A131938: 2,5,10,16,23,32,42,53,...
> >> Positive integers not in A131938: 1,3,4,6,7,8,9,11,...
> >> So A131937(8) = A131937(7) + 11 = 49.
> >> %Y A131937 A131938
> >> %O A131937 1
> >> %K A131937 ,more,nonn,
> >
> >> %I A131938
> >> %S A131938 2,5,10,16,23,32,42,53,65,78,93,109
> >> %N A131938 a(1)=2; a(2)=5. a(n) = a(n-1) + (nth positive integer which
> >> does not occur in sequence A131937).
> >> %e A131938 A131937: 1,4,8,14,21,29,...
> >> Positive integers not in A131937: 2,3,5,6,7,9,10,11,...
> >> So A131938(8) = A131938(7) + 11 = 53.
> >> %Y A131938 A131937
> >> %O A131938 1
> >> %K A131938 ,more,nonn,

[...]

> Could you please explain in detail, how you've got your sequences?

This is my PARI/GP code (for the first 50 terms) that confirms Leroy's values:

{
A=Set([1,4]); B=Set([2,5]);
a=4; na=4;
b=5; nb=3;
for(n=3,50,
  until(!setsearch(A,na),na++);
  until(!setsearch(B,nb),nb++);
  a+=nb;
  b+=na;
  A=setunion(A,[a]);
  B=setunion(B,[b]);
);
print(vecsort(eval(A)));
print(vecsort(eval(B)));
}

Here the sets A and B collect elements (as computed) of A131937 and
A131938 respectively; the variables a and b go over elements of A and
B; variables na and nb go over non-elements of A and B.

The output is:

[1, 4, 8, 14, 21, 29, 38, 49, 61, 74, 88, 103, 120, 138, 157, 177,
198, 220, 244, 269, 295, 322, 350, 379, 409, 440, 473, 507, 542, 578,
615, 653, 692, 732, 773, 816, 860, 905, 951, 998, 1046, 1095, 1145,
1196, 1248, 1302, 1357, 1413, 1470, 1528]

[2, 5, 10, 16, 23, 32, 42, 53, 65, 78, 93, 109, 126, 144, 163, 183,
205, 228, 252, 277, 303, 330, 358, 388, 419, 451, 484, 518, 553, 589,
626, 665, 705, 746, 788, 831, 875, 920, 966, 1013, 1061, 1111, 1162,
1214, 1267, 1321, 1376, 1432, 1489, 1547]

Regards,
Max



A018892 ("Number of ways to write 1/n as a sum of exactly 2 unit fractions")
has a simple formula a(n) = (d(n^2) + 1)/2, as noted in the first comment.

There is a simple construction not mentioned there that nicely demonstrates
the formula: 1/n = 1/(n+a) + 1/(n+b) implies ab = n^2.

The same construction extends to the general case: to write m/n as a sum
of 2 unit fractions, find a factorisation n^2 = xy, such that
n + x == n + y == 0 (mod m). Then m/n = m/(n+x) + m/(n+y), and the RHS

When m = 2, it remains simple: for odd n, any pair of factors of n^2 will
both be odd, so n+x, n+y will be even in every case. So given:

b(n) = Number of ways to write 2/n as a sum of exactly 2 unit fractions
we get:
b(n) = { A018892(n/2)  if n == 0 (mod 2)

For c(n) = Number of ways to write 3/n as a sum of exactly 2 unit fractions,
we need to find the number of ways to split n^2 into 2 factors each
equivalent either to 1 (mod 3) when n == 2 (mod 3), or vice versa.

Writing n = PQ, such that P is a product of primes == 1 (mod 3) and
Q a product of primes == 2 (mod 3), I find:

c(n) = { A018892(n/3)               if n == 0 (mod 3)

I have an approach to prove this, by induction on multiplication of prime
powers, but I suspect it is overly complex: can anyone suggest a simple
proof of c(n)?

I think the 2/n and 3/n cases are interesting enough to submit once I've
for other m, I don't expect to submit those unless they prove unexpectedly
interesting.

Hugo




From hv at crypt.org  Tue Jul 31 14:01:16 2007
From: hv at crypt.org (hv at crypt.org)
Date: Tue, 31 Jul 2007 13:01:16 +0100
Subject: A018892 and extensions
Message-ID: <200707311201.l6VC1GwJ004230@zen.crypt.org>

simplifies to two unit fractions.
       { A018892(n)    if n == 1 (mod 2)
       { d(P^2) (d(Q^2) - 1) / 4    if n == 1 (mod 3)
       { d(P^2) (d(Q^2) + 1) / 4    if n == 2 (mod 3)
satisfied myself of their correctness; while I plan to investigate m/n
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Max Alekseyev schrieb:
> On 7/28/07, Peter Pein <petsie at dordos.net> wrote:
> 
>>> Each here line contains a pair:
>>> n k
>>> such that sigma(3,k)=sigma(3,n) and k=a(n) for r=3 (following your
>>> notations above).
>>>
>>> I will let my program to run for a couple more days to reach 10^8 bound.
> 
> I stopped the program after it reached 7*10^7 as it started to eat too
> much memory.
> The results are attached. Please format them and submit to OEIS as you
> feel appropriate.
> 
> P.S. Eventually I've defended Ph.D. in Computer Science ;)

... if you felt that this has been necessary... ;-)
> 
> Regards,
> Max
> 
> 
Thank you again, Max.

As soon as I've got a little bit more time to spare, I'll enter the values

Peter




From petsie at dordos.net  Tue Jul 31 15:18:16 2007
From: petsie at dordos.net (Peter Pein)
Date: Tue, 31 Jul 2007 15:18:16 +0200
Subject: Do any integers occur in both sequences?
In-Reply-To: <46AEA87E.1010709@dordos.net>
References: <E1IFWBz-0000Pz-00@pop03.mail.atl.earthlink.net> <46AEA87E.1010709@dordos.net>
Message-ID: <46AF3698.6080307@dordos.net>

Well, I should not post as early in the morning (at least not before
I've got the third cup of coffee). I did not add a(n-1)....

Sorry for any inconvenience, trouble and the like

Peter

P.S.:
A131938-A131937 is (seems to be after coffee ;) ):

{1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6,
  7, 8, 8, 8, 8, 8, 8, 9, 10, 11, 11, 11, 11,
  11, 11, 12, 13, 14, 15, 15, 15, 15, 15, 15,
  15, 16, 17, 18, 19, 19, 19, 19, 19, 19, 19,
  19, 20, 21, 22, 23, 23, 23, 23, 23, 23, 23,
  23, 23, 24, 25, 26, 27, 27, 27, 27, 27, 27,
  27, 27, 27, 27, 28, 29, 30, 31, 32, 32, 32,
  32, 32, 32, 32, 32, 32, 32, 33, 34, 35, 36,
  37, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38,
  38, 39, 40, 41, 42, 43, 44, 44, 44, 44, 44,
  44, 44, 44, 44, 44, 44, 44, 45, 46, 47, 48,
  49, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50,
  50, 50, 50, 51, 52, 53, 54, 55, 56, 56, 56,
  56, 56}

Peter Pein schrieb:
> Leroy Quet schrieb:
>> I have just submitted these two interdependent sequences (So don't look 
>> for them in the database yet):
>>
>>> %I A131937
>>> %S A131937 1,4,8,14,21,29,38,49,61
>>> %N A131937 a(1)=1; a(2)=4. a(n) = a(n-1) + (nth positive integer which 
>>> does not occur in sequence A131938).
>>> %e A131937 A131938: 2,5,10,16,23,32,42,53,...
>>> Positive integers not in A131938: 1,3,4,6,7,8,9,11,...
>>> So A131937(8) = A131937(7) + 11 = 49.
>>> %Y A131937 A131938
>>> %O A131937 1
>>> %K A131937 ,more,nonn,
>>> %I A131938
>>> %S A131938 2,5,10,16,23,32,42,53,65,78,93,109
>>> %N A131938 a(1)=2; a(2)=5. a(n) = a(n-1) + (nth positive integer which 
>>> does not occur in sequence A131937).
>>> %e A131938 A131937: 1,4,8,14,21,29,...
>>> Positive integers not in A131937: 2,3,5,6,7,9,10,11,...
>>> So A131938(8) = A131938(7) + 11 = 53.
>>> %Y A131938 A131937
>>> %O A131938 1
>>> %K A131938 ,more,nonn,
>>
>> I have not thought about this too hard; so for all I know, the proof is 
>> quite easy.
>>
>> Do any positive integers occur in both A131937 and A131938?
>>
>> Thanks,
>> Leroy Quet
>>
> Hi again,
> 
> even if you meant "n-th pos. int. which does yet not occur in either l1
> or l2" (or does one say "..does yet neither occur in l1 nor in l2"?) I
> get another result
> 
> list1={1,4};list2={2,5};n=3;
> Do[
>     Print[{list1,list2}];
>     Print["n= ",n," ",Complement[Range[25],list2]];
>     AppendTo[list1,Part[Complement[Range[3n ],Union[list1,list2]],n]];
>     Print["n= ",n," ",Complement[Range[25],list1]];
>     AppendTo[list2,Part[Complement[Range[3n],Union[list1,list2]],n]];
>     n++;
>     ,{5}];
> list1
> list2
> Intersection[%%,%]
> 
....



I wrote:
>I have just submitted these two interdependent sequences (So don't look 
>for them in the database yet):
>
>>%I A131937
>>%S A131937 1,4,8,14,21,29,38,49,61
>>%N A131937 a(1)=1; a(2)=4. a(n) = a(n-1) + (nth positive integer which 
>>does not occur in sequence A131938).
>>%e A131937 A131938: 2,5,10,16,23,32,42,53,...
>>Positive integers not in A131938: 1,3,4,6,7,8,9,11,...
>>So A131937(8) = A131937(7) + 11 = 49.
>>%Y A131937 A131938
>>%O A131937 1
>>%K A131937 ,more,nonn,
>
>>%I A131938
>>%S A131938 2,5,10,16,23,32,42,53,65,78,93,109
>>%N A131938 a(1)=2; a(2)=5. a(n) = a(n-1) + (nth positive integer which 
>>does not occur in sequence A131937).
>>%e A131938 A131937: 1,4,8,14,21,29,...
>>Positive integers not in A131937: 2,3,5,6,7,9,10,11,...
>>So A131938(8) = A131938(7) + 11 = 53.
>>%Y A131938 A131937
>>%O A131938 1
>>%K A131938 ,more,nonn,
>
>I have not thought about this too hard; so for all I know, the proof is 
>quite easy.
>
>Do any positive integers occur in both A131937 and A131938?
>
>Thanks,
>Leroy Quet

I have made no progress towards determining if any particular positive 
integers occur within both sequences. (In other words: Does  A131937(k) = 
A131938(j) for any j and k {j and k are >= 1}, where j need not equal k?)

I conjecture that no particular positive integer occurs in both sequences.


Here is a smaller result related to these sequences, which I doubt will 
help (dis)prove the main conjecture:

Let a(n) =  A131937(n), b(n) = A131938(n), a(0) = b(0) = 0.

Let n = any positive integer.

Then n occurs (a(n) - a(n-1) - 1) times in sequence {b(n) - b(n-1) - n + 
1}.

And n occurs (b(n) - b(n-1) - 1) times in sequence {a(n) - a(n-1) - n + 
1}.

Not too earth-shattering -- but while we're on the subject...


Also, I wonder if anyone can come up with a closed form for {A131937(n)} 
and {A131938(n)}.
They seem like they might be related to Beautty sequences somehow.

Thanks,
Leroy Quet



A comment with seq A001006
http://www.research.att.com/~njas/sequences/A001006

This seems to be wrong, I get the counts

Note the starts math, Motzkins start as


Someone please verify!





From qq-quet at mindspring.com  Tue Jul 31 16:18:48 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Tue, 31 Jul 07 08:18:48 -0600
Subject: Do any integers occur in both sequences?
Message-ID: <E1IFsbM-0006LJ-00@pop05.mail.atl.earthlink.net>

says (near top):
  Number of sequences of length n-1 consisting of positive integers
  such that the opening and ending elements are 1 or 2, and the
  absolute difference between any 2 consecutive elements is 0 or 1.
 A024537 ,1, 2, 4, 9, 21, 50, 120, 289, 697, 1682 (=?= A018905)
     [1,] 1, 2, 4, 9, 21, 51, 127, 323,
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Date: Tue, 31 Jul 2007 10:50:49 -0700
From: "Max Alekseyev" <maxale at gmail.com>
To: "Joerg Arndt" <arndt at jjj.de>
Subject: Re: possibly wrong comment with Motzkin numbers
Cc: seqfan at ext.jussieu.fr
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On 7/31/07, Joerg Arndt <arndt at jjj.de> wrote:
> A comment with seq A001006
> http://www.research.att.com/~njas/sequences/A001006
> says (near top):
>   Number of sequences of length n-1 consisting of positive integers
>   such that the opening and ending elements are 1 or 2, and the
>   absolute difference between any 2 consecutive elements is 0 or 1.
>
> This seems to be wrong, I get the counts
>  A024537 ,1, 2, 4, 9, 21, 50, 120, 289, 697, 1682 (=?= A018905)
>
> Note the starts math, Motzkins start as
>      [1,] 1, 2, 4, 9, 21, 51, 127, 323,

With a simple recurrence formula implementation in PARI/GP, I confirm
that the comment in A001006 is correct:

{ a(n,l) = if(n==1,(l==1)||(l==2),if(l<=0,0,a(n-1,l-1)+a(n-1,l)+a(n-1,l+1))) }

{ f(n) = a(n,1)+a(n,2) }

? vector(10,n,f(n))
%1 = [2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798]

Here a(n,l) counts the number of sequences of positive integers of
length n, starting with 1 or 2 and ending with l.

Max





From diana.mecum at gmail.com  Sun Jul  1 05:58:30 2007
From: diana.mecum at gmail.com (Diana Mecum)
Date: Sat, 30 Jun 2007 22:58:30 -0500
Subject: Question related to sequence A066452
Message-ID: <a851af150706302058k5519146fr4bdd069236671ae5@mail.gmail.com>

Folks,

I am trying to add extension terms for sequence A066452. I have a question.

This is the internal text describing the sequence:

%I A066452
%S A066452
1,1,2,1,4,1,4,4,3,2,8,3,7,7,9,2,8,5,10,10,8,6,19,6,12,9,9,8,22,9,12,
%T A066452 12,15,10,31,9,11,14,24,13,23,9,24,17,16,10,35,15,23
%N A066452 Anti-phi(n).
%H A066452 Jon Perry, <a href="
http://www.users.globalnet.co.uk/~perry/maths/antidivisorother2.htm">
               Anti-phi function</a>
%F A066452 anti-phi(n) = number of integers <= n that are coprime to the
anti-divisors of n
%e A066452 10 has the anti-divisors 3,4,7. Therefore numbers coprime to
3,4,7 and less than
               10 are are 1,2,5, therefore anti-phi(10)=3.
%Y A066452 Cf. A058838, A066241.
%Y A066452 Sequence in context: A024994 A051953 A079277 this_sequence
A007104 A102627 A088296
%Y A066452 Adjacent sequences: A066449 A066450 A066451 this_sequence A066453
A066454 A066455
%K A066452 nonn,more,easy
%O A066452 2,3
%A A066452 Jon Perry (perry(AT)globalnet.co.uk), Dec 29 2001

I found a definition for "anti-divisor" as follows:

"Non-divisor: a number k which does not divide a given number x."
"Anti-divisor: a non-divisor k of x with the property that k is an odd
divisor of 2x-1 or 2x+1, or an even divisor of 2x."

I see how Jon gets 3, 4 and 7 as anti-divisors of 10. However, 2 is not
coprime to the anti-divisors of 10. He has 1, 2, and 5 as on the anti-phi
list.

The sequence which I derived for this sequence is:

1, 1, 2, 1, 4, 1, 4, 4, 3, 2, 2, 5, 3, 5, 4, 9, 2, 4, 5, 6, 6, 6, 6, 10, 5,
8, 6, 5, 8, 8, 9, 12,
 7, 10, 7, 12, 9, 8, 9, 13, 13, 9, 9, 14, 10

Can someone tell me if I am misunderstanding the definition of the sequence,
or if I have found an error?

Thanks,

Diana M.

-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From joshua.zucker at gmail.com  Sun Jul  1 06:44:41 2007
From: joshua.zucker at gmail.com (Joshua Zucker)
Date: Sat, 30 Jun 2007 21:44:41 -0700
Subject: Question related to sequence A066452
In-Reply-To: <a851af150706302058k5519146fr4bdd069236671ae5@mail.gmail.com>
References: <a851af150706302058k5519146fr4bdd069236671ae5@mail.gmail.com>
Message-ID: <721e81490706302144v4a71e477jbca73f5e29577eea@mail.gmail.com>

I don't understand it either, but at least I could use the wayback
machine to track down the URL given, at
  http://tinyurl.com/2xuvgr

And it says
  The anti-phi function is defined as the numbers <n that do not have
any anti-divisor as a factor.

Which may or may not be what they actually mean ... but at least it's
another possible interpretation to try.

--Joshua Zucker


On 6/30/07, Diana Mecum <diana.mecum at gmail.com> wrote:
> Folks,
>
> I am trying to add extension terms for sequence A066452. I have a question.
>
> This is the internal text describing the sequence:
>
> %I A066452
> %S A066452
> 1,1,2,1,4,1,4,4,3,2,8,3,7,7,9,2,8,5,10,10,8,6,19,6,12,9,9,8,22,9,12,
> %T A066452
> 12,15,10,31,9,11,14,24,13,23,9,24,17,16,10,35,15,23
> %N A066452 Anti-phi(n).
> %H A066452 Jon Perry, <a
> href="http://www.users.globalnet.co.uk/~perry/maths/antidivisorother2.htm
> ">
>                Anti-phi function</a>
> %F A066452 anti-phi(n) = number of integers <= n that are coprime to the
> anti-divisors of n
> %e A066452 10 has the anti-divisors 3,4,7. Therefore numbers coprime to
> 3,4,7 and less than
>                10 are are 1,2,5, therefore anti-phi(10)=3.
> %Y A066452 Cf. A058838, A066241.
> %Y A066452 Sequence in context: A024994 A051953 A079277 this_sequence
> A007104 A102627 A088296
> %Y A066452 Adjacent sequences: A066449 A066450 A066451 this_sequence A066453
> A066454 A066455
> %K A066452 nonn,more,easy
> %O A066452 2,3
> %A A066452 Jon Perry (perry(AT)globalnet.co.uk), Dec 29 2001
>
> I found a definition for "anti-divisor" as follows:
>
> "Non-divisor: a number k which does not divide a given number x."
> "Anti-divisor: a non-divisor k of x with the property that k is an odd
> divisor of 2x-1 or 2x+1, or an even divisor of 2x."
>
> I see how Jon gets 3, 4 and 7 as anti-divisors of 10. However, 2 is not
> coprime to the anti-divisors of 10. He has 1, 2, and 5 as on the anti-phi
> list.
>
> The sequence which I derived for this sequence is:
>
> 1, 1, 2, 1, 4, 1, 4, 4, 3, 2, 2, 5, 3, 5, 4, 9, 2, 4, 5, 6, 6, 6, 6, 10, 5,
> 8, 6, 5, 8, 8, 9, 12,
>  7, 10, 7, 12, 9, 8, 9, 13, 13, 9, 9, 14, 10
>
> Can someone tell me if I am misunderstanding the definition of the sequence,
> or if I have found an error?
>
> Thanks,
>
> Diana M.
>
> --
> "God made the integers, all else is the work of man."
> L. Kronecker, Jahresber. DMV 2, S. 19.




From zbi74583 at boat.zero.ad.jp  Sun Jul  1 06:57:40 2007
From: zbi74583 at boat.zero.ad.jp (koh)
Date: Sun, 01 Jul 2007 13:57:40 +0900
Subject: Quiz
Message-ID: <20070701045738.zbi74583@boat.zero.ad.jp>

    Hi, Seqfans

    What are these sequences?

    S1 : 1,1,1,0,1,0,1,-1,0,1,2,2....
    S2 : 1,2,3,-2,0,3,1,2,1,1....

       Hint.... Graph, Euler Number, S1=English, S2=Japanese

    Yasutoshi
    




From reismann at free.fr  Sun Jul  1 11:52:37 2007
From: reismann at free.fr (reismann at free.fr)
Date: Sun, 01 Jul 2007 11:52:37 +0200
Subject: =?iso-8859-1?b?Qe5vbg==?= and Chronos
Message-ID: <1183283557.468779659d8ca@imp.free.fr>

Hi seqfans,

A?on and Chronos form the Deleuze's concept of time :
Chronos : pulsated time (beats of our heart), the time of the history, the time
which passes.
A?on : the pure moment of time, the time of the events, of the "ecc??t?s", a
time at the same time too late and too early.

Ordinal and cardinal function of the numbers :
There are apples on the table.
1, 2, 3, 4, 5, 6. I count apples, the numbers have an ordinal function, the
action to count is held in the Chronos.
There are thus 6 apples. 6 take a cardinal function, we are not located more in
the Chronos, we are in another time : the A?on.

In our vision of the natural numbers, in the vision of the fundamental theorem
of the arithmetic, in the decomposition in weight*level with jump=0, we are in
the A?on.
To replace the natural numbers in the Chronos, it is enough to consider the
decomposition in weight*level+jump with jump=1.
We did not see the difference until now because the two decompositions give the
same result (with a different offset). I think that it is thus because the jump
is constant, the pulsation is regular.

In A?on, the prime numbers are primes, they are of level 1, it is all.
How to see in a new way prime numbers ?
By considering them in the Chronos, by analyzing them by the decomposition in
weight*level+jump.
In this case the jump is not constant, the pulsation is irregular.
There are "prime numbers" and "multiples" among the prime numbers ("prime
numbers" and "multiples" remain to be defined in this case).

The numbers have only one A?on but several Chronos. It is what I wanted to say
by ?Numbers are nothings?. The numbers are nothing if one does not specify in
which time one is located. Is the number in the A?on or the Chronos ? And if it
is in the Chronos, in which Chronos is it ?
11 in A?on : 11 is prime, is of level 1.
11 in the Chronos of natural numbers : 11 has a weight of 2, 11 = 2*5+1.
11 in the Chronos of prime numbers : 11 has a weight of 3 : 11 = 3*3+2.

Good thoughts,

R?mi Eismann

Deleuze on Wikipedia en :
http://en.wikipedia.org/wiki/Deleuze
Deleuze on Wikipedia fr :
http://fr.wikipedia.org/wiki/Gilles_Deleuze
On A?on and chronos (in french) :
Le vocabulaire de Deleuze (r?alis? par Rapha?l Bessis) -
http://tuxcafe.org/~renee/textes/deleuze/vocabulaire_deleuze.pdf





From diana.mecum at gmail.com  Sun Jul  1 16:56:15 2007
From: diana.mecum at gmail.com (Diana Mecum)
Date: Sun, 1 Jul 2007 09:56:15 -0500
Subject: Question related to sequence A066452
In-Reply-To: <721e81490706302144v4a71e477jbca73f5e29577eea@mail.gmail.com>
References: <a851af150706302058k5519146fr4bdd069236671ae5@mail.gmail.com>
	 <721e81490706302144v4a71e477jbca73f5e29577eea@mail.gmail.com>
Message-ID: <a851af150707010756o381941eau786ba39833a11510@mail.gmail.com>

Joshua,

Thanks a bunch. I would never have been able to find this different spin on
the definition of anti-phi.

I have been able to replicate the original list.

Diana M.

On 6/30/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
>
> I don't understand it either, but at least I could use the wayback
> machine to track down the URL given, at
>   http://tinyurl.com/2xuvgr
>
> And it says
>   The anti-phi function is defined as the numbers <n that do not have
> any anti-divisor as a factor.
>
> Which may or may not be what they actually mean ... but at least it's
> another possible interpretation to try.
>
> --Joshua Zucker
>
>
> On 6/30/07, Diana Mecum <diana.mecum at gmail.com> wrote:
> > Folks,
> >
> > I am trying to add extension terms for sequence A066452. I have a
> question.
> >
> > This is the internal text describing the sequence:
> >
> > %I A066452
> > %S A066452
> > 1,1,2,1,4,1,4,4,3,2,8,3,7,7,9,2,8,5,10,10,8,6,19,6,12,9,9,8,22,9,12,
> > %T A066452
> > 12,15,10,31,9,11,14,24,13,23,9,24,17,16,10,35,15,23
> > %N A066452 Anti-phi(n).
> > %H A066452 Jon Perry, <a
> > href="
> http://www.users.globalnet.co.uk/~perry/maths/antidivisorother2.htm
> > ">
> >                Anti-phi function</a>
> > %F A066452 anti-phi(n) = number of integers <= n that are coprime to the
> > anti-divisors of n
> > %e A066452 10 has the anti-divisors 3,4,7. Therefore numbers coprime to
> > 3,4,7 and less than
> >                10 are are 1,2,5, therefore anti-phi(10)=3.
> > %Y A066452 Cf. A058838, A066241.
> > %Y A066452 Sequence in context: A024994 A051953 A079277 this_sequence
> > A007104 A102627 A088296
> > %Y A066452 Adjacent sequences: A066449 A066450 A066451 this_sequence
> A066453
> > A066454 A066455
> > %K A066452 nonn,more,easy
> > %O A066452 2,3
> > %A A066452 Jon Perry (perry(AT)globalnet.co.uk), Dec 29 2001
> >
> > I found a definition for "anti-divisor" as follows:
> >
> > "Non-divisor: a number k which does not divide a given number x."
> > "Anti-divisor: a non-divisor k of x with the property that k is an odd
> > divisor of 2x-1 or 2x+1, or an even divisor of 2x."
> >
> > I see how Jon gets 3, 4 and 7 as anti-divisors of 10. However, 2 is not
> > coprime to the anti-divisors of 10. He has 1, 2, and 5 as on the
> anti-phi
> > list.
> >
> > The sequence which I derived for this sequence is:
> >
> > 1, 1, 2, 1, 4, 1, 4, 4, 3, 2, 2, 5, 3, 5, 4, 9, 2, 4, 5, 6, 6, 6, 6, 10,
> 5,
> > 8, 6, 5, 8, 8, 9, 12,
> >  7, 10, 7, 12, 9, 8, 9, 13, 13, 9, 9, 14, 10
> >
> > Can someone tell me if I am misunderstanding the definition of the
> sequence,
> > or if I have found an error?
> >
> > Thanks,
> >
> > Diana M.
> >
> > --
> > "God made the integers, all else is the work of man."
> > L. Kronecker, Jahresber. DMV 2, S. 19.
>



-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From jvospost3 at gmail.com  Sun Jul  1 18:28:56 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Sun, 1 Jul 2007 09:28:56 -0700
Subject: =?WINDOWS-1252?Q?Re:_A=EEon_and_Chronos?=
In-Reply-To: <1183283557.468779659d8ca@imp.free.fr>
References: <1183283557.468779659d8ca@imp.free.fr>
Message-ID: <5542af940707010928h4890e4b4ofa519b20a082694@mail.gmail.com>

Drifting away from seqfans primary function as I understand it, but
directly on your topic, as I understand it:

There is profane time, and there is sacred time. According to Eliade,
myths describe a time that is fundamentally different from historical
time (what modern man would consider "normal" time). "In short," says
Eliade, "myths describe ? breakthroughs of the sacred (or the
'supernatural') into the World".[7] The mythical age is the time when
the Sacred entered our world, giving it form and meaning: "The
manifestation of the sacred ontologically founds the world".[8] Thus,
the mythical age is sacred time, the only time that has value for
traditional man.

[7] Mircea Eliade, Myth and Reality, pg. 6
[8] Mircea Eliade, The Sacred and the Profane, pg. 21

http://en.wikipedia.org/wiki/Eternal_return_(Eliade)





From reismann at free.fr  Sun Jul  1 19:05:19 2007
From: reismann at free.fr (reismann at free.fr)
Date: Sun, 01 Jul 2007 19:05:19 +0200
Subject: =?iso-8859-1?b?Qe5vbg==?= and Chronos
In-Reply-To: <5542af940707010928h4890e4b4ofa519b20a082694@mail.gmail.com>
References: <1183283557.468779659d8ca@imp.free.fr> <5542af940707010928h4890e4b4ofa519b20a082694@mail.gmail.com>
Message-ID: <1183309519.4687decf17576@imp.free.fr>

I agree, it is not the place to speak philosophy but as nobody speaks about my
mathematical work...

I do not agree on this concept of time. The sacred time, the mythical age, the
"jadis', the ?it was once? of the tales are not times. They are "ecc??t?s", they
make much more than to define a temporal framework, they create an environment,
an atmosphere. The time of these "ecc??t?s" is indeed A?on.

But the disctinction between A?on and Chronos is not the same one as between the
sacred time and the profane time. Not. A?on is the pure moment of time, it is
the point on the arrow of time, it is the time of a photography. In mathematics
and sciences in general, one uses A?on, one analyzes photographs. By putting the
numbers in their Chronos, their fitting, their otherness, we see them in another
manner.

With the decomposition of the prime numbers in weight*level+gap, I replace the
prime numbers in their Chronos.

Best,

R?mi Eismann




From jvospost3 at gmail.com  Sun Jul  1 19:18:16 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Sun, 1 Jul 2007 10:18:16 -0700
Subject: =?ISO-8859-1?Q?Re:_A=EEon_and_Chronos?=
In-Reply-To: <1183309519.4687decf17576@imp.free.fr>
References: <1183283557.468779659d8ca@imp.free.fr>
	 <5542af940707010928h4890e4b4ofa519b20a082694@mail.gmail.com>
	 <1183309519.4687decf17576@imp.free.fr>
Message-ID: <5542af940707011018g6dff5435w42ece630acbb0496@mail.gmail.com>

I speak of your mathematical work: "It appears self consistent,
correct, and, to me, interesting."

Regarding Time, I have taught several hundred adult students a course
on "Time Travel: Math, Physics, Fiction" using as textbook:

Time Machines: Time Travel in Physics, Metaphysics, and Science
Fiction, by Paul J. Nahin, Springer-Verlag New York, 1993.

There is greater clarity in studing primes, and asymptotic limits of
real functions, than in the Philosophy of Time, or, perhaps, any
Philosophy.  So I shall for some time restrict my seqfans comments to
Mathematics and Integers.

Classically:

"... But the position of these and similar authorities is made clear
by Boethius, who says (V De Consolatione prosa 6), "When some people
hear that Plato thought this world neither had a beginning in time nor
will ever have an end, they mistakenly conclude that the created world
is coeternal with the Creator. However, to be led through the endless
life Plato attributes to the world is one thing; to embrace
simultaneously the whole presence of endless life is quite another,
and it is this latter that is proper to the divine mind." [PL 63,
859B]

<a href="http://www.fordham.edu/halsall/basis/aquinas-eternity.html">Medieval
Sourcebook:
Thomas Aquinas:
On The Eternity of the World
(DE AETERNITATE MUNDI)
DE AETERNITATE MUNDI [[1]]
Translation (c) 1991, 1997 by Robert T. Miller[[2]] </a>



Since people are posting sequence puzzles on seq.fan lately, I thought I 
would post this sequence puzzle of a different varity.

I suspect this 'puzzle' is easy, and I'll probably regret I posted this.

---

Let {c(k)} be as defined at sequence A022940. ({c(k)} itself is not in 
the EIS.)

Define sequence {a(k)} as follows:

Let b(n) = c(n) - n + 1.

a(1) = the number of 1's in {b(k)}. a(2) = the number of 2's in {b(k)}.
In general, a(n) = the number of n's in {b(k)}.

So, {a(k)} begins: 0,1,1,3,5,6,7,9,...

Define {a(k)}. (Define it in a simpler way than by the steps given above.)

Thanks,
Leroy Quet




From qq-quet at mindspring.com  Mon Jul  2 00:42:07 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Sun, 1 Jul 07 16:42:07 -0600
Subject: Another kind of sequence puzzle
References: <1183283557.468779659d8ca@imp.free.fr>
Message-ID: <E1I58AC-0006iC-00@pop03.mail.atl.earthlink.net>

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

SeqFiends,

I can't recall the context, but we have several times over
the past few years been discussing several questions about
boolean functions B^k -> B and boolean mappings B^k -> B^n.

I am finally getting some of the work that I referred to
in slightly prettier shape.  Here's a link to a paper on
Differential Logic:

http://www.centiare.com/Differential_Logic_and_Dynamic_Systems

Related material can be found by perusing this directory page:

http://www.centiare.com/Directory:Jon_Awbrey

Some of this work even has a little bit to do with the
love that Eternity bears toward the creatures of Time.

Many Regards,

Jon Awbrey

CC: Arisbe Forum: http://stderr.org/pipermail/arisbe/
CC: Inquiry List: http://stderr.org/pipermail/inquiry/

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
inquiry e-lab: http://stderr.org/pipermail/inquiry/
?iare: http://www.centiare.com/Directory:Jon_Awbrey
getwiki: http://www.getwiki.net/-User_talk:Jon_Awbrey
zhongwen wp: http://zh.wikipedia.org/wiki/User:Jon_Awbrey
http://www.altheim.com/ceryle/wiki/Wiki.jsp?page=JonAwbrey
wp review: http://wikipediareview.com/index.php?showuser=398
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o





From Eric.Angelini at kntv.be  Mon Jul  2 17:28:54 2007
From: Eric.Angelini at kntv.be (Eric Angelini)
Date: Mon, 2 Jul 2007 17:28:54 +0200
Subject: Seq and first diff show the same "digit pattern"
Message-ID: <F05775148D000C4B9321A5113D53CB7D405D2B@KNTV-SERVER01.kntv.local>


Hello SeqFan,
could someone compute a few more terms of this seq:

1  12  14  155  160  211  271  292  419  548  572  691 ...

The principle is:

- Seq and first diff show the same "digit pattern".

S = 1  12  14   155  160  211  271  292   419   548  572  691 ...
d =  11   2  141    5   51   60   21   127   129   24   19 ...

Rules:
- start S with "1"
- add to the last term of S the smallest integer d no yet
  added and not present in S such that the concatenation
  of S's terms and the concatenation of all ds are the
  same string of digits

So, never twice the same integer in sequence or first
differences. 

I'm quite sure that all N's will be split between S and d.

Best,
?.

http://www.research.att.com/~njas/sequences/A110621  
has a close Mathematica pgm by Robert G. Wilson.

(thanks again to him!)




Something is wrong with my email, and I haven't been receiving all emails 
have.

I will post the solution now, even though only a short while has passed 

Original email as spoiler-space.

>Since people are posting sequence puzzles on seq.fan lately, I thought I 
>would post this sequence puzzle of a different varity.
>
>I suspect this 'puzzle' is easy, and I'll probably regret I posted this.
>
>---
>
>Let {c(k)} be as defined at sequence A022940. ({c(k)} itself is not in 
>the EIS.)
>
>Define sequence {a(k)} as follows:
>
>Let b(n) = c(n) - n + 1.
>
>a(1) = the number of 1's in {b(k)}. a(2) = the number of 2's in {b(k)}.
>In general, a(n) = the number of n's in {b(k)}.
>
>So, {a(k)} begins: 0,1,1,3,5,6,7,9,...
>
>Define {a(k)}. (Define it in a simpler way than by the steps given above.)
>
>Thanks,
>Leroy Quet


I get that a(1)=0, a(2)=1. a(n) = c(n-2) -1, for n >= 3.

I don't know if there is a "closed form" (nonrecursive representation) 
for {c(k)}.

Thanks,
leroy Quet




From qq-quet at mindspring.com  Mon Jul  2 19:29:58 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Mon, 2 Jul 07 11:29:58 -0600
Subject: Another kind of sequence puzzle
Message-ID: <E1I5Ple-0005xt-00@pop06.mail.atl.earthlink.net>

sent to me. So I don't know if someone has posted the solution before I 
since I posted the question.
Return-Path: <xordan.tom at gmail.com>
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Message-ID: <fe71ae2b0707021836s19391bdaqd911c68dd9b3f373 at mail.gmail.com>
Date: Mon, 2 Jul 2007 20:36:10 -0500
From: xordan <xordan.tom at gmail.com>
To: seqfan at ext.jussieu.fr
Subject: relatively primes and prime numbers graphs.
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I understand that the attached file and the folloging words that  don't
coincide with the strict discussion line in seqfan, but they contain some
graphic curiosities that I wanted to share with the members of the list. I
wait some benevolent comments. Again I notice that the translation of the
original in Spanish is made with the help of software:

The attached file (coprimes.zip ) contains one book of calculation sheets
that has 5 work sheets that give results (to my view) interesting related
with the numbers relatively primes.
The first sheet shows the numbers (1 to 256) relatively primes to each
other  that added (arithmetic sum)   becomes  a prime number as  result; the
second sheet shows the numbers relatively primes  whose absolute difference
becomes a prime number  as  result; the third are the conjunction of the
previous two , that is to say the numbers relatively primes  whose their
algebraic sum  gives as  result a prime  number. The fourth is the same
graph that it appears in the current page
http://mathworld.wolfram.com/RelativelyPrime.html (RelativelyPrime.gif) and
that it shows the primes numbers relatively to each other. The fifth work
sheet is the conjunction of RelativelyPrime.gif and the previous sheet
"abs(r+-c)=prime". - With this it is shown graphically that all the numbers
(r,c) whose algebraic sum is a prime number (p) they are relatively prime to
each other.
IF  r+c=p  THEN   coprime(r,c)=1.-
-- 
xordan at hotmail.com
xordan_co at yahoo.com
xordan.tom at gmail.com

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<div>I understand that the attached file and the folloging words that&nbsp; don&#39;t coincide with the strict discussion line in seqfan, but they contain some graphic curiosities that&nbsp;I wanted to share with the members of the list. I wait some benevolent comments. Again I notice that the translation of the original in Spanish is made with the help of software: 
</div>
<div>&nbsp;</div>
<div>The attached file (coprimes.zip )&nbsp;contains&nbsp;one book of calculation sheets that has 5 work&nbsp;sheets that give results (to my view) interesting related with the numbers relatively primes.&nbsp;&nbsp; <br>The first sheet shows the numbers (1&nbsp;to 256) relatively primes to each other&nbsp; that added (arithmetic sum)&nbsp;&nbsp; becomes&nbsp; a prime number as&nbsp; result; the second sheet shows the numbers relatively primes&nbsp; whose absolute difference becomes a prime number&nbsp; as&nbsp; result; the third are the&nbsp;conjunction of the previous two , that is to say the numbers relatively primes&nbsp; whose their&nbsp; algebraic sum&nbsp; gives as&nbsp; result a prime&nbsp; number. The fourth is the same graph that it appears in the current page 
<a href="http://mathworld.wolfram.com/RelativelyPrime.html">http://mathworld.wolfram.com/RelativelyPrime.html</a>&nbsp;(RelativelyPrime.gif) and that it shows the primes numbers relatively to each other. The fifth work sheet is the conjunction of 
RelativelyPrime.gif and the previous sheet &quot;abs(r+-c)=prime&quot;. - With this it is shown graphically that all the numbers (r,c) whose algebraic sum is a prime number (p) they are relatively prime to each other.</div>

<div>IF&nbsp;&nbsp;r+c=p&nbsp;&nbsp;THEN&nbsp;&nbsp;&nbsp;coprime(r,c)=1.- <br>-- <br><a href="mailto:xordan at hotmail.com">xordan at hotmail.com</a><br><a href="mailto:xordan_co at yahoo.com">xordan_co at yahoo.com</a><br><a href="mailto:xordan.tom at gmail.com">xordan.tom at gmail.com
</a> </div>

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TMTqxc71ln1Q+Yyh7qH4J2/ZC/6Atze91Zu8fcRHv9n7OmPmNV3oxC2/iOdt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------=_Part_104140_14761177.1183426570634--



Maximilian Hasler has pointed out to me that at sequence A022940 
"complement" is not defined.

I assume that Clark Kimberling intended {c(k)} to be those positive 
integers (in order) that do not appear in A022940.

So {c(k)} begins: 2,4,6,7,8,10,11,12,13,14,16,17,...

Thanks,
Leroy Quet



Hello Seqfans,

I ordered 2 sequence numbers. I didn't receive the confirmation email and on the page with numbers there is the following paragraph:

Insufficient disk space; try again later Insufficient disk space; try again later returntosender: cannot select queue for oeis Insufficient disk space; try again later returntosender: cannot select queue for postmaster putbody: write error: No space left on device Error writing control file qfl63ErU4Y019338: No space left on device 

Is OEIS working?
Can I use my numbers?

Tanya


_________________________________________________________________
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From qq-quet at mindspring.com  Tue Jul  3 17:09:05 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Tue, 3 Jul 07 09:09:05 -0600
Subject: Another kind of sequence puzzle
In-Reply-To: <721e81490707022117o386e3adau95c08683f9d9aa35@mail.gmail.com>
References: <fe71ae2b0707021836s19391bdaqd911c68dd9b3f373@mail.gmail.com>
	 <721e81490707022117o386e3adau95c08683f9d9aa35@mail.gmail.com>
Message-ID: <E1I5k2r-0006Ep-00@pop04.mail.atl.earthlink.net>

------=_Part_114770_16554780.1183484165194
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Content-Transfer-Encoding: 7bit
Content-Disposition: inline

Yes Joshua:
The only exception ocurrs if r | c or if c | r in the case abs(r-c)=prime;
in other words if r and c are multiples of p.


2007/7/2, Joshua Zucker <joshua.zucker at gmail.com>:
>
> Proof: if r and c are not coprime, then they have a common divisor d,
> and then d divides r+c and r-c as well.
>
> The only possible exceptions would be cases like 6 and 3, where 6-3 is
> prime even though 6 and 3 are not coprime.  That is, where the common
> divisor is a prime equal to r-c.
>
> --Joshua Zucker
>
>
> On 7/2/07, xordan <xordan.tom at gmail.com> wrote:
> > I understand that the attached file and the folloging words that  don't
> > coincide with the strict discussion line in seqfan, but they contain
> some
> > graphic curiosities that I wanted to share with the members of the list.
> I
> > wait some benevolent comments. Again I notice that the translation of
> the
> > original in Spanish is made with the help of software:
> >
> > The attached file (coprimes.zip ) contains one book of calculation
> sheets
> > that has 5 work sheets that give results (to my view) interesting
> related
> > with the numbers relatively primes.
> > The first sheet shows the numbers (1 to 256) relatively primes to each
> other
> >  that added (arithmetic sum)   becomes  a prime number as  result; the
> > second sheet shows the numbers relatively primes  whose absolute
> difference
> > becomes a prime number  as  result; the third are the conjunction of the
> > previous two , that is to say the numbers relatively primes  whose their
> > algebraic sum  gives as  result a prime  number. The fourth is the same
> > graph that it appears in the current page
> > http://mathworld.wolfram.com/RelativelyPrime.html
> > (RelativelyPrime.gif) and that it shows the primes numbers relatively to
> > each other. The fifth work sheet is the conjunction of
> RelativelyPrime.gif
> > and the previous sheet "abs(r+-c)=prime". - With this it is shown
> > graphically that all the numbers (r,c) whose algebraic sum is a prime
> number
> > (p) they are relatively prime to each other.
> > IF  r+c=p  THEN   coprime(r,c)=1.-
> > --
> > xordan at hotmail.com
> > xordan_co at yahoo.com
> > xordan.tom at gmail.com
> >
>



-- 
xordan at hotmail.com
xordan_co at yahoo.com
xordan.tom at gmail.com

------=_Part_114770_16554780.1183484165194
Content-Type: text/html; charset=ISO-8859-1
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<div>Yes Joshua:</div>
<div>The only exception ocurrs if r | c&nbsp;or if c | r in the case abs(r-c)=prime; in other words if r and c&nbsp;are multiples of p.&nbsp;</div>
<div>&nbsp;<br>&nbsp;</div>
<div><span class="gmail_quote">2007/7/2, Joshua Zucker &lt;<a href="mailto:joshua.zucker at gmail.com">joshua.zucker at gmail.com</a>&gt;:</span>
<blockquote class="gmail_quote" style="PADDING-LEFT: 1ex; MARGIN: 0px 0px 0px 0.8ex; BORDER-LEFT: #ccc 1px solid">Proof: if r and c are not coprime, then they have a common divisor d,<br>and then d divides r+c and r-c as well.
<br><br>The only possible exceptions would be cases like 6 and 3, where 6-3 is<br>prime even though 6 and 3 are not coprime.&nbsp;&nbsp;That is, where the common<br>divisor is a prime equal to r-c.<br><br>--Joshua Zucker<br><br><br>
On 7/2/07, xordan &lt;<a href="mailto:xordan.tom at gmail.com">xordan.tom at gmail.com</a>&gt; wrote:<br>&gt; I understand that the attached file and the folloging words that&nbsp;&nbsp;don&#39;t<br>&gt; coincide with the strict discussion line in seqfan, but they contain some
<br>&gt; graphic curiosities that I wanted to share with the members of the list. I<br>&gt; wait some benevolent comments. Again I notice that the translation of the<br>&gt; original in Spanish is made with the help of software:
<br>&gt;<br>&gt; The attached file (coprimes.zip ) contains one book of calculation sheets<br>&gt; that has 5 work sheets that give results (to my view) interesting related<br>&gt; with the numbers relatively primes.<br>&gt; The first sheet shows the numbers (1 to 256) relatively primes to each other
<br>&gt;&nbsp;&nbsp;that added (arithmetic sum)&nbsp;&nbsp; becomes&nbsp;&nbsp;a prime number as&nbsp;&nbsp;result; the<br>&gt; second sheet shows the numbers relatively primes&nbsp;&nbsp;whose absolute difference<br>&gt; becomes a prime number&nbsp;&nbsp;as&nbsp;&nbsp;result; the third are the conjunction of the
<br>&gt; previous two , that is to say the numbers relatively primes&nbsp;&nbsp;whose their<br>&gt; algebraic sum&nbsp;&nbsp;gives as&nbsp;&nbsp;result a prime&nbsp;&nbsp;number. The fourth is the same<br>&gt; graph that it appears in the current page<br>&gt; <a href="http://mathworld.wolfram.com/RelativelyPrime.html">
http://mathworld.wolfram.com/RelativelyPrime.html</a><br>&gt; (RelativelyPrime.gif) and that it shows the primes numbers relatively to<br>&gt; each other. The fifth work sheet is the conjunction of RelativelyPrime.gif<br>
&gt; and the previous sheet &quot;abs(r+-c)=prime&quot;. - With this it is shown<br>&gt; graphically that all the numbers (r,c) whose algebraic sum is a prime number<br>&gt; (p) they are relatively prime to each other.<br>
&gt; IF&nbsp;&nbsp;r+c=p&nbsp;&nbsp;THEN&nbsp;&nbsp; coprime(r,c)=1.-<br>&gt; --<br>&gt; <a href="mailto:xordan at hotmail.com">xordan at hotmail.com</a><br>&gt; <a href="mailto:xordan_co at yahoo.com">xordan_co at yahoo.com</a><br>&gt; <a href="mailto:xordan.tom at gmail.com">
xordan.tom at gmail.com</a><br>&gt;<br></blockquote></div><br><br clear="all"><br>-- <br><a href="mailto:xordan at hotmail.com">xordan at hotmail.com</a><br><a href="mailto:xordan_co at yahoo.com">xordan_co at yahoo.com</a><br><a href="mailto:xordan.tom at gmail.com">
xordan.tom at gmail.com</a> 

------=_Part_114770_16554780.1183484165194--



Are other people also having problems receiving the automated reply when 
with this.)

I guess we will have to wait until Neil gets back from vacation to submit 
via this method.

Thanks,
Leroy Quet






From qq-quet at mindspring.com  Wed Jul  4 18:16:42 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Wed, 4 Jul 07 10:16:42 -0600
Subject: No automatic replies
Message-ID: <E1I67Zp-0002UH-00@pop05.mail.atl.earthlink.net>

submitting sequences via the on-line form? (I know Tanya has had problems 
Return-Path: <jvospost3 at gmail.com>
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Message-ID: <5542af940707041259v435e0f9alf913531e86f7230c at mail.gmail.com>
Date: Wed, 4 Jul 2007 12:59:45 -0700
From: "Jonathan Post" <jvospost3 at gmail.com>
To: "Leroy Quet" <qq-quet at mindspring.com>,
   "jonathan post" <jvospost2 at yahoo.com>
Subject: Re: No automatic replies
Cc: seqfan at ext.jussieu.fr
In-Reply-To: <E1I67Zp-0002UH-00 at pop05.mail.atl.earthlink.net>
MIME-Version: 1.0
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X-Greylist: IP, sender and recipient auto-whitelisted, not delayed by milter-greylist-3.0 (shiva.jussieu.fr [134.157.0.166]); Wed, 04 Jul 2007 21:59:47 +0200 (CEST)
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X-Miltered: at shiva.jussieu.fr with ID 468BFC32.001 by Joe's j-chkmail (http://j-chkmail.ensmp.fr)!

I got no autoreply for this:

NEW SEQUENCE FROM Jonathan Vos Post

%I A000001
%S A000001 -109, -42, -26, -18, -14, -12, -10, -9, -8,
-7, -6, -6, -5, -5, -5, -4, -4, -4, -4, -4, -3, -3,
-3, -3, -3, -3, -3, -3, -2, -2, -2, -2, -2, -2, -2,
-2, -2, -2, -2, -2, -2, -2, -2, -1
%N A000001 Floor of constants in De Bruijn's approach
to weighted Carleman's inequality.
%C A000001 Abstract of Gao: "We study finite sections
of weighted Carleman's inequality following the
approach of De Bruijn. Similar to the unweighted case,
we obtain an asymptotic expression for the optimal
constant."
%D A000001 N. G. De Bruijn, Carleman's inequality for
finite series, Nederl. Akad. Wetensch. Proc. Ser. A 66
= Indag, pp. 505-514.
%H A000001 Peng Gao, <a
href="http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0077v1.pdf">Finite
Sections of Weighted Carleman's Inequality</a>, 30
June 2007, p.1.
%F A000001 a(n) = floor(e -
(2*(pi^2)*e)/((log(n))^2)).
%e A000001 a(2) = -109 because e -
(2*(pi^2)*e)/((log(2))^2) ~
-108.9611770171388392925257212314455433803548032218666994709.
a(3) = -42 because e - (2*(pi^2)*e)/((log(3))^2) ~
-41.7382232411477828847325690963577817095329948893743754723.
a(4) = -26 because e - (2*(pi^2)*e)/((log(4))^2) ~
-25.20158288294042589661121470434688897177076548519170518650.
a(30) = -2 because e - (2*(pi^2)*e)/((log(30))^2) ~
-1.92003649778404604739381818236913112747520.
a(45) = -1 because e - (2*(pi^2)*e)/((log(45))^2) ~
-0.98456269963010489451493724472555817336322761419762175593.
%O A000001 2,1
%K A000001 ,easy,sign,
%A A000001 Jonathan Vos Post (jvospost2 at yahoo.com),
Jul 03 2007




From jvospost3 at gmail.com  Wed Jul  4 22:01:04 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Wed, 4 Jul 2007 13:01:04 -0700
Subject: No automatic replies
In-Reply-To: <5542af940707041259v435e0f9alf913531e86f7230c@mail.gmail.com>
References: <E1I67Zp-0002UH-00@pop05.mail.atl.earthlink.net>
	 <5542af940707041259v435e0f9alf913531e86f7230c@mail.gmail.com>
Message-ID: <5542af940707041301s32f82ca9h2f1e3638a33ddecc@mail.gmail.com>

Nor did I get an autoreply for this:

COMMENT FROM Jonathan Vos Post RE A122505

%I A122505
%S A122505 24, 24, 95, 1, 143, 1, 262, -213, 453,
-261, 739, -833, 1169, -1168, 2172, -2505, 3104,
-3581, 5255, -6449
%N A122505 Arises from energy spectrum of three
dimensional gravity with negative cosmological
constant, in analysis by Edward Witten.
%C A122505 a(11)-a(20) given by "GH" at Not Even Wrong
Blog; not verified as correct; commenter email
unknown.
%H A122505 "GH", <a
href="http://www.math.columbia.edu/~woit/wordpress/?p=571#comment-26500">Strings
2007</a>, thread of "Not Even Wrong" Physics blog,
edited by Peter Woit, <woit at math.columbia.edu>, thread
initiated 25 June 2007, comment by "GH" extending
sequence dated 1 July 2007.
%O A122505 1
%K A122505 ,sign,
%A A122505 Jonathan Vos Post (jvospost2 at yahoo.com),
Jul 03 2007




From jvospost3 at gmail.com  Wed Jul  4 22:05:27 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Wed, 4 Jul 2007 13:05:27 -0700
Subject: No automatic replies
In-Reply-To: <5542af940707041301s32f82ca9h2f1e3638a33ddecc@mail.gmail.com>
References: <E1I67Zp-0002UH-00@pop05.mail.atl.earthlink.net>
	 <5542af940707041259v435e0f9alf913531e86f7230c@mail.gmail.com>
	 <5542af940707041301s32f82ca9h2f1e3638a33ddecc@mail.gmail.com>
Message-ID: <5542af940707041305r34b54aa2w219a86bc732a103a@mail.gmail.com>

For that matter (and I'm not saying that ANY of these are good or
important, just that they track to OEIS and the Literature):

COMMENT FROM Jonathan Vos Post RE A001399

%I A001399
%S A001399 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30,
33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108,
114, 120, 127, 133, 140, 147, 154, 161, 169, 176, 184, 192, 200, 208,
217, 225, 234, 243, 252, 261, 271, 280, 290, 300, 310, 320, 331, 341
%N A001399 Number of partitions of n into at most 3 parts; also
partitions of n+3 in which the greatest part is 3; also multigraphs
with 3 nodes and n edges.
%H A001399 Andrew N. Norris, <a
href="http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0115v1.pdf">
Higher derivatives and the inverse derivative of a tensor-valued
function of a tensor</a>, 1 July 2007, Equation 3.28, p.10
%F A001399 After initial 1 appears identical to integer part of
((n+4)^2 + 4)/12, which is given Norris as the number of points in,
and on the boundary of the integer grid of {I, J}, bounded by the
three straight lines I = 0, I - J = 0, and I + 2J = n + 1.
%O A001399 1
%K A001399 ,nonn,
%A A001399 Jonathan Vos Post (jvospost2 at yahoo.com), Jul 03 2007




From martin_n_fuller at btinternet.com  Fri Jul  6 21:34:13 2007
From: martin_n_fuller at btinternet.com (Martin Fuller)
Date: Fri, 6 Jul 2007 20:34:13 +0100 (BST)
Subject: Help correcting A003294 and A096739
Message-ID: <324718.39838.qm@web86601.mail.ird.yahoo.com>

SeqFans,

A003294 and A096739 have some obvious errors, but they may also be the same
sequence.  Can anyone help provide terms and/or decide whether they are
duplicates?

http://www.research.att.com/~njas/sequences/?q=id:A003294|id:A096739

The first error is that there are missing multiples, e.g. 1765 = 353*5.  Then
there is a typo in A096739 which has 5129 instead of 5729.  It looks like the
values have been brought together from several sources but not checked
carefully enough.

Are there any fourth power sums which involve repeated values?  If the answer
is proved to be no, then these sequences can be combined.

Martin Fuller




From maximilian.hasler at gmail.com  Sat Jul  7 00:33:01 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Fri, 6 Jul 2007 18:33:01 -0400
Subject: Help correcting A003294 and A096739
In-Reply-To: <324718.39838.qm@web86601.mail.ird.yahoo.com>
References: <324718.39838.qm@web86601.mail.ird.yahoo.com>
Message-ID: <3c3af2330707061533m3328ccb0vec62ccd85c7db644@mail.gmail.com>

> http://www.research.att.com/~njas/sequences/?q=id:A003294|id:A096739
>
> The first error is that there are missing multiples, e.g. 1765 = 353*5.

I think this is an omission in the definition rather than error in the
sequence -
I believe in general for such "homogeneous" diophantine equations
it is somehow tacitly understood that multiples are not listed as solutions
(since any multiple of a solution (A,B,C,D,n) is trivially again a solution,
one only gives solutions with gcd(A,B,C,D,n) = 1.)

but of course this should be somehow said in the definition.

M.H.




From jvospost3 at gmail.com  Sat Jul  7 00:33:48 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Fri, 6 Jul 2007 15:33:48 -0700
Subject: Correction to, comment on, A061789, and primes therein
Message-ID: <5542af940707061533l7a0654a7u2ef96454c2cae8db@mail.gmail.com>

Correction to A061789:

a(4) = 2^2 + 3^3 + 5^5 + 7^7 = 826699
rather than, as shown, 826696.

Comment:

Primes in A061689

a(2) = 2^2 + 3^3 = 31 is prime

a(4) = 2^2 + 3^3 + 5^5 + 7^7 = 826699 is prime

a(24) = 2^2 + 3^3 + 5^5 + 7^7 + 11^11 + 13^13 + 17^17 + 19^19
+ 23^23 + 29^29 + 31^31 + 37^37 + 41^41 + 43^43 +
47^47 + 53^53 + 59^59 + 61^61 + 67^67 + 71^71 + 73^73
+ 79^79 + 83^83 + 89^89 =
313119 843606 266222 723931 946804 611259 287100
533754 507329 247768 328563 715123 287679 948437
264517 293404 303260 795541 448480 275059 086587
161357 892405 477908 284669 463639 709544 036228
659629 is prime

A061789 Sum_{k=1..n} p(k)^p(k), p(k) = k-th prime (A000040).
       4, 31, 3156, 826696, 285312497310, 303160419089563,
827240565046755853740, 1979246896225360344977719,
20880469979094808259715377888286,
2567686153182091604540923022990731504371755




From djr at nk.ca  Sat Jul  7 04:46:18 2007
From: djr at nk.ca (Don Reble)
Date: Fri, 06 Jul 2007 20:46:18 -0600
Subject: Help correcting A003294 and A096739
In-Reply-To: <324718.39838.qm@web86601.mail.ird.yahoo.com>
References: <324718.39838.qm@web86601.mail.ird.yahoo.com>
Message-ID: <468EFE7A.1040400@nk.ca>

Seqfans:

    A003294 goes
         353  651  706 1059 1302 1412 1765 1953 2118 2471
        2487 2501 2604 2824 2829 3177 3255 3530 3723 3883
        3906 3973 4236 4267 4333 4449 4557 4589 4942 4949
        4974 5002 5208 5281 5295 5463 5491 5543 5648 5658
        5729 5859 6001 6167 6354 6510 6609 6707 6801 7060
        7101 7161 7209 7339 7413 7446 7461 7503 7703 7766
        7812 7946 8119 8373 8433 8463 8472 8487 8493 8517
        8534 8577 8637 8666 8825 8898 9114 9137 9178 9243
        9431 9519 9531 9639 9765 9797 9877 9884 9898 9948
    and all of these are in A096739. So they still might be the same.

    One might make a sequence of the primitive terms, where the fourth
    roots have no common factor. Ah! that's the intent of A039664.
    Dr. Sloane, do you have the original author?

    I'll send A003294 and A039664 edits to Dr. Sloane.

-- 
Don Reble  djr at nk.ca



-- 
This message has been scanned for viruses and
dangerous content by MailScanner, and is
believed to be clean.





From jvospost3 at gmail.com  Mon Jul  9 00:31:49 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Sun, 8 Jul 2007 15:31:49 -0700
Subject: Correction to, comment on, A061789, and primes therein
In-Reply-To: <5542af940707061533l7a0654a7u2ef96454c2cae8db@mail.gmail.com>
References: <5542af940707061533l7a0654a7u2ef96454c2cae8db@mail.gmail.com>
Message-ID: <5542af940707081531o25ddaca4g5019982b3f02d3d@mail.gmail.com>

I've verified, using Alpertron, and double-checking with the WIMS
server at U.Nice, that there are no more primes in A061789 through
a(78):

2^2 + 3^3 + 5^5 + 7^7 + ... + 397^397

which has 1032 digits.

hence, if there is a 4th prime in A061789 it must be a Titanic prime,
to use the term coined by Samuel Yates in the 1980s, denoting a prime
number of at least 1000 decimal digits.




From jvospost3 at gmail.com  Tue Jul 10 02:03:30 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Mon, 9 Jul 2007 17:03:30 -0700
Subject: Prime Fermat number-twins (Boris V. Tarasov, arXiv)
Message-ID: <5542af940707091703r69cc1411h385ccd4a7919158e@mail.gmail.com>

Is it worth submitting this sequence?

Prime Fermat number-twins.

5, 7, 13, 19, 65539

Prime numbers of the form (2^(2^n)) - 3 or (2^(2^n)) + 3

The concrete theory of numbers : Problem of simplicity of Fermat number-twins
http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0907v1.pdf
Authors: Boris V. Tarasov
Comments: 6 pages
Subjects: General Mathematics (math.GM)

The problem of simplicity of Fermat number-twins f_n^{plus or minus) =
2^(2^n){plus or minus 3} is studied. The question for what n numbers
f_n^{plus or minus) are composite is investigated. The
factor-identities for numbers of a kind x^2 {plus or minus} k $ are
found.

a(n) are (sorted) from Tarasov, p.5

If this is worth submitting, what is the next value a(5) asserted by
Tarasov to be equal or greater than (2^(2^17)) - 3?




From jvospost3 at gmail.com  Tue Jul 10 02:10:47 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Mon, 9 Jul 2007 17:10:47 -0700
Subject: Marek Wolf, arXiv: Analog of the Skewes number for twin primes
Message-ID: <5542af940707091710o27abe8ft727a16b924a95413@mail.gmail.com>

Dear Marek Wolf,

I certainly hope that you submit sequence(s) to the Online
Encyclopedia of IUnteger Sequences from your fine paper:

arXiv:0707.0980
    Title: Analog of the Skewes number for twin primes
    Authors: Marek Wolf
    Subjects: Number Theory (math.NT)
6 July 2007

Sincerely,

Prof. Jonathan Vos Post




From zakseidov at yahoo.com  Tue Jul 10 15:57:55 2007
From: zakseidov at yahoo.com (zak seidov)
Date: Tue, 10 Jul 2007 06:57:55 -0700 (PDT)
Subject: Concise Search Result
In-Reply-To: <5542af940707041305r34b54aa2w219a86bc732a103a@mail.gmail.com>
Message-ID: <93596.85791.qm@web38208.mail.mud.yahoo.com>

Dear seqfans,

I think that Concise Search Result
of the following form would be nice, right?
Zak
 
Search: 18, 96, 600, 4320
Result:
A001563, A052633,  A052655, A094258, A094304, A109074


       
____________________________________________________________________________________
Need a vacation? Get great deals
to amazing places on Yahoo! Travel.
http://travel.yahoo.com/




From ogerard at ext.jussieu.fr  Tue Jul 10 18:42:39 2007
From: ogerard at ext.jussieu.fr (=?ISO-8859-1?Q?Olivier_G=E9rard?=)
Date: Tue, 10 Jul 2007 18:42:39 +0200
Subject: [seqfan] INTERRUPTION DE SERVICE - MAILING LIST HIATUS
Message-ID: <4693B6FF.3010103@ext.jussieu.fr>



Dear list members,


The machine hosting the seqfan list server will be shut down from

Thursday July 12 6pm GMT+2    until    Friday July 13 9am GMT+2

(that's the end of tuesday morning till 1 or 2am friday US East time)



Please do not send any message during this period as you will
not be able to check whether it has been received or not by
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I will send a mail to the list so that everyone knows operations
are back to normal.  Thanks in advance for your patience and
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with my best regards,

Olivier GERARD

PS: As usual, any inquiries about this matter should be directed
to me (olivier.gerard at gmail.com or olivier.gerard at paris7.jussieu.fr)
and not to the list.

===============================
===============================

Chers abonn?s de la liste seqfan

Arr?t temporaire du serveur h?bergeant la liste

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jusqu'au

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S'il vous plait, n'envoyez pas de message pendant cette p?riode,
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J'enverrai un courier ? la liste quand la situation sera revenue
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Merci d'avance pour votre patience et votre discipline.


Cordialement,

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PS: Comme d'habitude, si vous avez des questions ? ce sujet, ?crivez
moi directement (olivier.gerard at gmail.com ou 
olivier.gerard at paris7.jussieu.fr ) et pas ? la liste.







From diana.mecum at gmail.com  Wed Jul 11 15:22:53 2007
From: diana.mecum at gmail.com (Diana Mecum)
Date: Wed, 11 Jul 2007 08:22:53 -0500
Subject: Question related to sequence A071267
Message-ID: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>

Sequence Fans,

I am looking at sequence A071267.

%I A071267
%S A071267
2,4,6,8,10,11,12,14,16,18,22,33,44,55,66,77,88,99,101,110,121,132,143,
%T A071267 154,165,176,187,198,111,222,666,888,1110
%N A071267 Numbers which can be expressed as the sum of all distinct digit
permutations of
               some number k.
%C A071267 222 can be expressed so in two different ways i.e. 222= 200 +020
+ 002 as well as
               222= 101 +110 +011. Question: find a number which can be so
expressed
               in n different ways.
%e A071267 1110 is a member as a sum of all distinct permutations of 104.
i.e. 104,140,410,
               401,014,041.
%Y A071267 Sequence in context: A081472 A097660 A067030 this_sequence
A072427 A050420 A096922
%Y A071267 Adjacent sequences: A071264 A071265 A071266 this_sequence A071268
A071269 A071270
%K A071267 base,more,nonn
%O A071267 1,1
%A A071267 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 01 2002

When I try to follow the rule for generating the sequence numbers, I get the
following list;

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
121, 132, 143, 154, 165,
 176, 187, 222, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443, 1554,
1665, 1776, 1887,
 1998, 2109, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996

Can someone explain why my list differs from the original? I am not
understanding the hypothesis to generate the original list.

Diana M.

-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From joshua.zucker at gmail.com  Wed Jul 11 17:12:36 2007
From: joshua.zucker at gmail.com (Joshua Zucker)
Date: Wed, 11 Jul 2007 08:12:36 -0700
Subject: Question related to sequence A071267
In-Reply-To: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>
References: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>
Message-ID: <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com>

Hi Diana and all,
Murthy's sequences often have errors.  Here, among other problems, the
definition is "numbers which ..." but then shouldn't they be SORTED?

I don't quite agree with your list, either, though.  Here's what I
think, based purely on brute-force with a computer, using as "seeds"
all the numbers up to 100000:

1 2 3 4 5 6 7 8 9 11 22 33 44 55 66 77 88 99 110 111 121 132 143 154
165 176 187 222 333 444 555 666 777 888 999 1110 1111 1221 1332 1443
1554 1665 1776 1887 1998 2109 2220 2222 2331 2442 2553 2664 2775 2886
3108 3330 3333 3552 3774 3996 4218 4440 4444 4662 4884 5106 5328 5555
6666 7777 8888 9999 11110 11111 12221 13332 14443 15554 16665 17776
18887 19998 21109 22220 22222 23331 24442 25553 26664 27775 28886
29997

which leaves me wondering, for instance, why I got 29997 but not 2997
in there.  But now I see it's a nice little bit of combinatorics: 108
goes to 1998 (six permutations, so effectively each spot is
(1+0+8+1+0+8) so we get 1800 + 180 + 18), while 1008 goes to 29997
(twelve permutations, so each spot is (1+0+0+8+1+0+0+8+1+0+0+8) so we
get 27000 + 2700 + 270 + 27).

So I think my above list of terms are correct.  Differences from your
list, Diana, are that I have 333 and 1111 and 2222 and 2331 and 2553
and 2775 and 3333 which are missing from your list.  (2331 for example
comes from 399 -> 399 + 939 + 993, and any list of identical digits
maps to itself, so 333 -> 333.)

Diana, would you verify that you agree with my list of terms and then
if you do please submit the corrected terms for this sequence?

Thanks,
--Joshua Zucker


On 7/11/07, Diana Mecum <diana.mecum at gmail.com> wrote:
> Sequence Fans,
>
> I am looking at sequence A071267.
>
> %I A071267
> %S A071267
> 2,4,6,8,10,11,12,14,16,18,22,33,44,55,66,77,88,99,101,110,121,132,143,
> %T A071267 154,165,176,187,198,111,222,666,888,1110
> %N A071267 Numbers which can be expressed as the sum of all distinct digit
> permutations of
>                some number k.
> %C A071267 222 can be expressed so in two different ways i.e. 222= 200 +020
> + 002 as well as
>                222= 101 +110 +011. Question: find a number which can be so
> expressed
>                 in n different ways.
> %e A071267 1110 is a member as a sum of all distinct permutations of 104.
> i.e. 104,140,410,
>                401,014,041.
> %Y A071267 Sequence in context: A081472 A097660 A067030 this_sequence
> A072427 A050420 A096922
> %Y A071267 Adjacent sequences: A071264 A071265 A071266 this_sequence A071268
> A071269 A071270
> %K A071267 base,more,nonn
> %O A071267 1,1
> %A A071267 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 01 2002
>
> When I try to follow the rule for generating the sequence numbers, I get the
> following list;
>
> 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
> 121, 132, 143, 154, 165,
>  176, 187, 222, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443, 1554,
> 1665, 1776, 1887,
>  1998, 2109, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996
>
> Can someone explain why my list differs from the original? I am not
> understanding the hypothesis to generate the original list.
>
> Diana M.
>
> --
> "God made the integers, all else is the work of man."
> L. Kronecker, Jahresber. DMV 2, S. 19.




From g.resta at iit.cnr.it  Wed Jul 11 17:11:28 2007
From: g.resta at iit.cnr.it (Giovanni Resta)
Date: Wed, 11 Jul 2007 17:11:28 +0200
Subject: Question related to sequence A071267
In-Reply-To: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>
References: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>
Message-ID: <4694F320.3010100@iit.cnr.it>

Diana Mecum wrote:
> %I A071267
> %S A071267
> 2,4,6,8,10,11,12,14,16,18,22,33,44,55,66,77,88,99,101,110,121,132,143,
> %T A071267 154,165,176,187,198,111,222,666,888,1110
> %N A071267 Numbers which can be expressed as the sum of all distinct
> digit permutations of
>                some number k.
> When I try to follow the rule for generating the sequence numbers, I
> get the following list;
>
> 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110,
> 111, 121, 132, 143, 154, 165,
>  176, 187, 222, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443,
> 1554, 1665, 1776, 1887,
>  1998, 2109, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996
>
> Can someone explain why my list differs from the original? I am not
> understanding the hypothesis to generate the original list.
First of all, you must bear in mind that a consistent fraction of the
sequences submitted by Amarnath Murthy
are erroneous.

In any case it seems to me that also your list is not consistent with
the definition.
In particular, considering elements less or equal to 3996, you list
lacks the following
terms: 333, 1111, 2222, 2331, 2553, 2775, 3333
It is clear that all the repdigits (like 333, 1111, 2222, 3333, and so
on) belong to the
sequence since, for example, the set of all the distinct permutations of
333 is just 333.
(moreover 333 is also produced by the permutations of 300: 300+030+003 =
333).
The other missing values are produced as :
2331 = 399 + 993 + 939
2553 = 599+995+959
2775 = 799+997+979

So I think that the correct list, up to 10000, is
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
121, 132, 143, 154, 165, 176, 187, 222, 333, 444, 555, 666, 777, 888, 999,
1110, 1111, 1221, 1332, 1443, 1554, 1665, 1776, 1887, 1998, 2109, 2220,
2222,
2331, 2442, 2553, 2664, 2775, 2886, 3108, 3330, 3333, 3552, 3774, 3996,
4218,
4440, 4444, 4662, 4884, 5106, 5328, 5555, 6666, 7777, 8888, 9999

while, if we let drop the 'distinct' clause (so for example 11 produces
not 11 but 11+11=22), the
elements  up to 10000 are
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121,
132,
143, 154, 165, 176, 187, 198, 222, 444, 666, 888, 1110, 1332, 1554, 1776,
1998, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996, 4218, 4440,
4662,
4884, 5106, 5328, 5550, 5772, 5994, 6666

g.





From diana.mecum at gmail.com  Wed Jul 11 18:40:30 2007
From: diana.mecum at gmail.com (Diana Mecum)
Date: Wed, 11 Jul 2007 09:40:30 -0700
Subject: Question related to sequence A071267
In-Reply-To: <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com>
References: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>
	 <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com>
Message-ID: <a851af150707110940q2d6c38c1wf292f3ebe55e962@mail.gmail.com>

Joshua and Giovanni,

Thanks. I hadn't done a comprehensive initial list, having only looked at
200 terms. My question was more directed to why 1, 3, 5, 7 and 9 were not on
the list.

I will assume that my algorithm is correct, which agrees with your findings.

Thank you,

Diana M.


On 7/11/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
>
> Hi Diana and all,
> Murthy's sequences often have errors.  Here, among other problems, the
> definition is "numbers which ..." but then shouldn't they be SORTED?
>
> I don't quite agree with your list, either, though.  Here's what I
> think, based purely on brute-force with a computer, using as "seeds"
> all the numbers up to 100000:
>
> 1 2 3 4 5 6 7 8 9 11 22 33 44 55 66 77 88 99 110 111 121 132 143 154
> 165 176 187 222 333 444 555 666 777 888 999 1110 1111 1221 1332 1443
> 1554 1665 1776 1887 1998 2109 2220 2222 2331 2442 2553 2664 2775 2886
> 3108 3330 3333 3552 3774 3996 4218 4440 4444 4662 4884 5106 5328 5555
> 6666 7777 8888 9999 11110 11111 12221 13332 14443 15554 16665 17776
> 18887 19998 21109 22220 22222 23331 24442 25553 26664 27775 28886
> 29997
>
> which leaves me wondering, for instance, why I got 29997 but not 2997
> in there.  But now I see it's a nice little bit of combinatorics: 108
> goes to 1998 (six permutations, so effectively each spot is
> (1+0+8+1+0+8) so we get 1800 + 180 + 18), while 1008 goes to 29997
> (twelve permutations, so each spot is (1+0+0+8+1+0+0+8+1+0+0+8) so we
> get 27000 + 2700 + 270 + 27).
>
> So I think my above list of terms are correct.  Differences from your
> list, Diana, are that I have 333 and 1111 and 2222 and 2331 and 2553
> and 2775 and 3333 which are missing from your list.  (2331 for example
> comes from 399 -> 399 + 939 + 993, and any list of identical digits
> maps to itself, so 333 -> 333.)
>
> Diana, would you verify that you agree with my list of terms and then
> if you do please submit the corrected terms for this sequence?
>
> Thanks,
> --Joshua Zucker
>
>
> On 7/11/07, Diana Mecum <diana.mecum at gmail.com> wrote:
> > Sequence Fans,
> >
> > I am looking at sequence A071267.
> >
> > %I A071267
> > %S A071267
> > 2,4,6,8,10,11,12,14,16,18,22,33,44,55,66,77,88,99,101,110,121,132,143,
> > %T A071267 154,165,176,187,198,111,222,666,888,1110
> > %N A071267 Numbers which can be expressed as the sum of all distinct
> digit
> > permutations of
> >                some number k.
> > %C A071267 222 can be expressed so in two different ways i.e. 222= 200
> +020
> > + 002 as well as
> >                222= 101 +110 +011. Question: find a number which can be
> so
> > expressed
> >                 in n different ways.
> > %e A071267 1110 is a member as a sum of all distinct permutations of
> 104.
> > i.e. 104,140,410,
> >                401,014,041.
> > %Y A071267 Sequence in context: A081472 A097660 A067030 this_sequence
> > A072427 A050420 A096922
> > %Y A071267 Adjacent sequences: A071264 A071265 A071266 this_sequence
> A071268
> > A071269 A071270
> > %K A071267 base,more,nonn
> > %O A071267 1,1
> > %A A071267 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 01 2002
> >
> > When I try to follow the rule for generating the sequence numbers, I get
> the
> > following list;
> >
> > 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
> > 121, 132, 143, 154, 165,
> >  176, 187, 222, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443,
> 1554,
> > 1665, 1776, 1887,
> >  1998, 2109, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996
> >
> > Can someone explain why my list differs from the original? I am not
> > understanding the hypothesis to generate the original list.
> >
> > Diana M.
> >
> > --
> > "God made the integers, all else is the work of man."
> > L. Kronecker, Jahresber. DMV 2, S. 19.
>



-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From diana.mecum at gmail.com  Wed Jul 11 19:29:42 2007
From: diana.mecum at gmail.com (Diana Mecum)
Date: Wed, 11 Jul 2007 10:29:42 -0700
Subject: Question related to sequence A071267
In-Reply-To: <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com>
References: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>
	 <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com>
Message-ID: <a851af150707111029i23b67dcbgf2950a71f4be5d2c@mail.gmail.com>

Joshua,

Yes, I will double check that your and my terms agree. I will submit the
corrected list, as that was my purpose in studying the list in the first
place.

Thanks for your time.

Diana M.


On 7/11/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
>
> Hi Diana and all,
> Murthy's sequences often have errors.  Here, among other problems, the
> definition is "numbers which ..." but then shouldn't they be SORTED?
>
> I don't quite agree with your list, either, though.  Here's what I
> think, based purely on brute-force with a computer, using as "seeds"
> all the numbers up to 100000:
>
> 1 2 3 4 5 6 7 8 9 11 22 33 44 55 66 77 88 99 110 111 121 132 143 154
> 165 176 187 222 333 444 555 666 777 888 999 1110 1111 1221 1332 1443
> 1554 1665 1776 1887 1998 2109 2220 2222 2331 2442 2553 2664 2775 2886
> 3108 3330 3333 3552 3774 3996 4218 4440 4444 4662 4884 5106 5328 5555
> 6666 7777 8888 9999 11110 11111 12221 13332 14443 15554 16665 17776
> 18887 19998 21109 22220 22222 23331 24442 25553 26664 27775 28886
> 29997
>
> which leaves me wondering, for instance, why I got 29997 but not 2997
> in there.  But now I see it's a nice little bit of combinatorics: 108
> goes to 1998 (six permutations, so effectively each spot is
> (1+0+8+1+0+8) so we get 1800 + 180 + 18), while 1008 goes to 29997
> (twelve permutations, so each spot is (1+0+0+8+1+0+0+8+1+0+0+8) so we
> get 27000 + 2700 + 270 + 27).
>
> So I think my above list of terms are correct.  Differences from your
> list, Diana, are that I have 333 and 1111 and 2222 and 2331 and 2553
> and 2775 and 3333 which are missing from your list.  (2331 for example
> comes from 399 -> 399 + 939 + 993, and any list of identical digits
> maps to itself, so 333 -> 333.)
>
> Diana, would you verify that you agree with my list of terms and then
> if you do please submit the corrected terms for this sequence?
>
> Thanks,
> --Joshua Zucker
>
>
> On 7/11/07, Diana Mecum <diana.mecum at gmail.com> wrote:
> > Sequence Fans,
> >
> > I am looking at sequence A071267.
> >
> > %I A071267
> > %S A071267
> > 2,4,6,8,10,11,12,14,16,18,22,33,44,55,66,77,88,99,101,110,121,132,143,
> > %T A071267 154,165,176,187,198,111,222,666,888,1110
> > %N A071267 Numbers which can be expressed as the sum of all distinct
> digit
> > permutations of
> >                some number k.
> > %C A071267 222 can be expressed so in two different ways i.e. 222= 200
> +020
> > + 002 as well as
> >                222= 101 +110 +011. Question: find a number which can be
> so
> > expressed
> >                 in n different ways.
> > %e A071267 1110 is a member as a sum of all distinct permutations of
> 104.
> > i.e. 104,140,410,
> >                401,014,041.
> > %Y A071267 Sequence in context: A081472 A097660 A067030 this_sequence
> > A072427 A050420 A096922
> > %Y A071267 Adjacent sequences: A071264 A071265 A071266 this_sequence
> A071268
> > A071269 A071270
> > %K A071267 base,more,nonn
> > %O A071267 1,1
> > %A A071267 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 01 2002
> >
> > When I try to follow the rule for generating the sequence numbers, I get
> the
> > following list;
> >
> > 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
> > 121, 132, 143, 154, 165,
> >  176, 187, 222, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443,
> 1554,
> > 1665, 1776, 1887,
> >  1998, 2109, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996
> >
> > Can someone explain why my list differs from the original? I am not
> > understanding the hypothesis to generate the original list.
> >
> > Diana M.
> >
> > --
> > "God made the integers, all else is the work of man."
> > L. Kronecker, Jahresber. DMV 2, S. 19.
>



-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From davidwwilson at comcast.net  Thu Jul 12 07:41:32 2007
From: davidwwilson at comcast.net (David Wilson)
Date: Thu, 12 Jul 2007 01:41:32 -0400
Subject: Question related to sequence A071267
References: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com> <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com> <a851af150707111029i23b67dcbgf2950a71f4be5d2c@mail.gmail.com>
Message-ID: <000f01c7c447$4deb5920$6501a8c0@yourxhtr8hvc4p>


Let f(n) be the sum of all permutes of n.

Let

    s(n) = sum of digits of n.
    d(n) = number of digits of n.
    c_n(k) = number of occurences of digit k in n.
    p(n) = PROD(k = 0..9; c_n(k)!).
    r(n) = n-digit rep-1 number = (10^n-1)/n.
    t(n) = ((s(n)(d(n)-1)!)/p(n)).

Then f(n) = t(n) r(d(n)).


For example, if n = 314159, we get

    s(n) = 23
    d(n) = 6
    c_n = (0, 2, 0, 1, 1, 1, 0, 0, 0, 1)
    p(n) = PROD(k = 0..9; c_n(k)!) = 2
    r(d(n)) = r(6) = 111111
    t(n) = (23*120)/2 = 1380

and

    f(314159) = 1380*11111 = 153333180


A combinatorial argument shows that when we add the permutes of n, the same number of each digit will appear in each column of the sum. This means that

    f(n) = (sum of each column) * (d(n)-digit rep-1 number)
         = (sum of each column) * r(d(n))

from which we conclude

    t(n) = sum of each column

implying that t(n) is integer and r(d(n)) | f(n).


Using this knowledge, we can explore why f(n) = 29997 is insoluble while f(n) = 2997 is not.

Let n have d(n) digits. Then n <= 9 r(d(n)) and has at most d(n)! permutes. This means that

    [1] f(n) <= 9 r(d(n)) d(n)!

Suppose f(n) = 29997. We know that r(d(n)) | f(n), and the only rep-1 numbers that divide 29997 are r(1) = 1, r(2) = 11 and r(4) = 1111, implying d(n) = 1, 2 or 4. From [1],

    d(n) = 1  =>  f(n) <= 9
    d(n) = 2  =>  f(n) <= 198
    d(n) = 4  =>  f(n) <= 239976

Given f(n) = 29997, we must have d(n) = 4 and r(d(n)) = 1111.

This means that

    t(n) = f(n)/r(d(n)) = 29997/1111 = 27.

so that

    ((s(n)(d(n)-1)!)/p(n)) = 27.

Since d(n) = 4, we have

    6 s(n)/p(n) = 27
    => s(n) = 9(p(n)/2).

Since p(n) and s(n) are both integer, p(n) must be even to make the right side integer. This means 9 | s(n) <= 9 d(n) = 36. This means that s(n) = 9, 18, 27 or 36.

Suppose s(n) = 9. Then p(n) = 2. By the definition of p(n), p(n) = 2 iff exactly one digit appears twice in n and all others once (or not at all). So we are looking for a 4-digit n with a single repeated digit and digit sum 9. Many such numbers exist, 1008 being the smallest, and indeed f(1008) = 29997.


But now look at f(n) = 2997.  Following similar logic to the above, we find that d(n) = 3 and t(n) = 27 giving

    ((s(n)(d(n)-1)!)/p(n)) = 27
    => 2 s(n)/p(n) = 27
    => s(n) = 27(p(n)/2).

Again, p(n) must be even, so 27 | s(n). But s(n) <= 9 d(n) = 27, so s(n) = 27 forcing n = 999 and f(n) = 999. Hence f(n) = 2997 is insoluble.




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From njas at research.att.com  Thu Jul 12 10:45:29 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Thu, 12 Jul 2007 04:45:29 -0400 (EDT)
Subject: 3 seqs that should be the same but are not
In-Reply-To: <200707120845.l6C8jTGh29153108@fry.research.att.com>
References: <200707120845.l6C8jTGh29153108@fry.research.att.com>
Message-ID: <200707120845.l6C8jTGh29153108@fry.research.att.com>

Why should "surface area" (A074457) equal "volume" (A087300)?




From DWCantrell at sigmaxi.net  Thu Jul 12 15:01:07 2007
From: DWCantrell at sigmaxi.net (David W. Cantrell)
Date: Thu, 12 Jul 2007 14:01:07 +0100
Subject: 3 seqs that should be the same but are not
References: <200707120845.l6C8jTGh29153108@fry.research.att.com> <5542af940707120512v585336e2vb67179da5f641008@mail.gmail.com>
Message-ID: <001501c7c484$b79a7a30$25933e44@Dell>

----- Original Message ----- 
From: "Jonathan Post" <jvospost3 at gmail.com>
To: <njas at research.att.com>
Cc: <seqfan at ext.jussieu.fr>; <eric at weisstein.com>; <rgwv at rgwv.com>; 
"jonathan post" <jvospost2 at yahoo.com>
Sent: Thursday, July 12, 2007 13:12
Subject: Re: 3 seqs that should be the same but are not


> Why should "surface area" (A074457) equal "volume" (A087300)?

They're not the same constant, but the constant given in A074457 is 
simply 2 more than the constant given in A087300. Both entries should 
mention that, I think.

There are also two errors in the title of A087300. It should read:
Decimal expansion of value of d for which volume of d-dimensional unit 
ball is maximized.
[Note that it is necessary to specify that it's a _unit_ ball.]

David 





From DWCantrell at sigmaxi.net  Thu Jul 12 16:05:33 2007
From: DWCantrell at sigmaxi.net (David W. Cantrell)
Date: Thu, 12 Jul 2007 15:05:33 +0100
Subject: 3 seqs that should be the same but are not
References: <181140.23791.qm@web86606.mail.ukl.yahoo.com>
Message-ID: <001001c7c48d$b81894c0$25933e44@Dell>

Martin,

Concerning your question "Can someone explain why the dimension with 
maximum surface area is exactly 2 more than the dimension with maximum 
volume?":

On a "mechanical" level, explaining why is trivial. If we let invdig 
denote the appropriate inverse of the digamma function, then we can 
solve the required transcendental equations to obtain

2 invdig(log(pi))

and

2 invdig(log(pi)) + 2.

But you were probably wanting something deeper, and I'm not sure what 
that is.

David

----- Original Message ----- 
From: "Martin Fuller" <martin_n_fuller at btinternet.com>
To: "David W. Cantrell" <DWCantrell at sigmaxi.net>; "Jonathan Post" 
<jvospost3 at gmail.com>; <njas at research.att.com>
Cc: <seqfan at ext.jussieu.fr>; <eric at weisstein.com>; <rgwv at rgwv.com>; 
"jonathan post" <jvospost2 at yahoo.com>
Sent: Thursday, July 12, 2007 14:53
Subject: Re: 3 seqs that should be the same but are not


> A074455 and A087300 should be the same: A074455 has the right 
> description and
> came first, A087300 has the right values.
>
> I get a different value to OEIS for A074457 as well as for A074455.
>
> See below for the current values and my calculations in PARI.  The 
> PARI code
> gives the same value as A087300 and is consistent when increasing 
> the
> precision, so I think it's right.
>
> Martin Fuller
>
> PS Can someone explain why the dimension with maximum surface area 
> is exactly 2
> more than the dimension with maximum volume?
>
> PPS All these constants are based on geometers dimension which is 
> one more than
> topologists dimension (see the comments at MathWorld and compare 
> with the
> formulae at PlanetMath
> http://planetmath.org/encyclopedia/VolumeOfTheNSphere.html)
>
> OEIS
> A087300
> 5.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926
> A074455
> 5.25694640486057678013283838869076923661901723718321485750987966544135040807732427416016036408330066793834
> A074457
> 7.25694640486057678013283838869076923661901723718321485750987966581295771725141939748631782716397233020121
> A074454
> 5.27776802111340099728214586417284638752999928451017356776163734021486412730547017110062048407258401284645
> A074456
> 33.1611944849620026918630240155829735800472328410872585131001181554037565464718434466607460949351387694776
>
> PARI/GP
> A074455
> 5.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926957
> A074457
> 7.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926957
> A074454
> 5.27776802111340099728214586417284638752999928451017356776163734021486412730547017110062048407258401284645
> A074456
> 33.1611944849620026918630240155829735800472328410872585131001181554037565464718434466607460949351387694776
>
> /* PARI/GP code */
> hyperspheresurface(d)=2*Pi^(d/2)/gamma(d/2)
> hyperspherevolume(d)=hyperspheresurface(d)/d
>
> FindMax(fn_x,lo,hi)=
> {
>  local(oldprecision, x, y, z);
>  oldprecision = default(realprecision);
>  default(realprecision, oldprecision+10);
>
>  while (hi-lo > 10^-oldprecision,
>    while (1,
>      z = vector(2, i, lo*(3-i)/3 + hi*i/3);
>      y = vector(2, i, eval(Str("x = z[" i "]; " fn_x)));
>      if (abs(y[1]-y[2]) > 10^(5-default(realprecision)), break);
>      default(realprecision, default(realprecision)+10);
>    );
>    if (y[1] < y[2], lo = z[1], hi = z[2]);
>  );
>
>  default(realprecision, oldprecision);
>  (lo + hi) / 2.
> }
>
> default(realprecision, 105);
> A074455=FindMax("hyperspherevolume(x)", 1, 9)
> A074457=FindMax("hyperspheresurface(x)", 1, 9)
> A074454=hyperspherevolume(A074455)
> A074456=hyperspheresurface(A074457)
> /* PARI/GP code ends */
>
> --- "David W. Cantrell" <DWCantrell at sigmaxi.net> wrote:
>
>> ----- Original Message ----- 
>> From: "Jonathan Post" <jvospost3 at gmail.com>
>> To: <njas at research.att.com>
>> Cc: <seqfan at ext.jussieu.fr>; <eric at weisstein.com>; <rgwv at rgwv.com>;
>> "jonathan post" <jvospost2 at yahoo.com>
>> Sent: Thursday, July 12, 2007 13:12
>> Subject: Re: 3 seqs that should be the same but are not
>>
>>
>> > Why should "surface area" (A074457) equal "volume" (A087300)?
>>
>> They're not the same constant, but the constant given in A074457 is
>> simply 2 more than the constant given in A087300. Both entries 
>> should
>> mention that, I think.
>>
>> There are also two errors in the title of A087300. It should read:
>> Decimal expansion of value of d for which volume of d-dimensional 
>> unit
>> ball is maximized.
>> [Note that it is necessary to specify that it's a _unit_ ball.]
>>
>> David
>>
>> 





From diana.mecum at gmail.com  Thu Jul 12 16:36:00 2007
From: diana.mecum at gmail.com (Diana Mecum)
Date: Thu, 12 Jul 2007 09:36:00 -0500
Subject: Question related to sequence A071267
In-Reply-To: <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com>
References: <a851af150707110622y11e1b5faob5525e9a4ac7795d@mail.gmail.com>
	 <721e81490707110812w7313fb79jba97b8181521661e@mail.gmail.com>
Message-ID: <a851af150707120736o7e035362t73e6c2a4bc08914e@mail.gmail.com>

Joshua,

Performing permutations on numbers up to 100,000 gave me the following
numbers to add to sequence A071267. I will add these to the sequence. Diana
M.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
121, 132, 143, 154, 165,
 176, 187, 222, 333, 444, 555, 666, 777, 888, 999, 1110, 1111, 1221, 1332,
1443, 1554, 1665, 1776,
 1887, 1998, 2109, 2220, 2222, 2331, 2442, 2553, 2664, 2775, 2886, 3108,
3330, 3333, 3552, 3774,
 3996, 4218, 4440, 4444, 4662, 4884, 5106, 5328, 5555, 6666, 7777, 8888,
9999, 11110, 11111,
 12221, 13332, 14443, 15554, 16665, 17776, 18887, 19998, 21109, 22220,
22222, 23331, 24442, 25553,
 26664, 27775, 28886, 29997, 31108, 32219, 33330, 33333, 34441, 35552,
36663, 37774, 38885, 39996,
 43329, 44444, 46662, 49995, 53328, 55555, 56661, 59994, 63327, 66660,
66666, 69993, 73326, 76659,
 77777, 79992, 83325, 86658, 88888, 89991, 93324, 96657, 99990, 99999,
103323, 106656, 109989,
 111110, 111111, 113322, 119988, 122221, 126654, 133320, 133332, 139986,
144443, 146652, 153318,
 155554, 159984, 166650, 166665, 173316, 177776, 179982, 186648, 188887,
193314, 199980, 199998,
 211109, 222220, 233331, 244442, 255553, 266664, 277775, 288886, 299997,
311108, 322219, 333330,
 344441, 355552, 366663, 377774, 388885, 399996, 411107, 422218, 433329,
444440, 455551, 466662,
 477773, 488884, 511106, 533328, 555550, 577772, 599994, 622216, 644438,
666660, 688882, 711104,
 733326, 755548, 777770, 799992, 822214, 844436, 866658, 888880, 911102,
933324, 955546, 977768,
 999990, 1022212, 1066656, 1111100, 1133322, 1155544, 1199988, 1244432,
1266654, 1288876, 1333320,
 1377764, 1399986, 1422208, 1466652, 1511096, 1533318, 1555540, 1599984,
1644428, 1666650,
 1688872, 1733316, 1777760, 1799982, 1822204, 1866648, 1933314, 1999980,
2066646, 2133312,
 2199978, 2266644, 2333310, 2399976, 2466642, 2533308, 2599974, 2666640,
2733306, 2799972,
 2933304, 3066636, 3199968, 3333300, 3466632, 3599964, 3733296, 3866628,
3999960, 4133292,
 4266624, 4399956, 4533288, 4666620, 4799952, 4933284, 5066616, 5199948,
5333280, 5599944,
 5866608, 6133272, 6399936, 6666600, 6933264, 7199928, 7466592, 7733256,
7999920, 8266584,
 8533248, 8799912, 9066576, 9333240

On 7/11/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
>
> Hi Diana and all,
> Murthy's sequences often have errors.  Here, among other problems, the
> definition is "numbers which ..." but then shouldn't they be SORTED?
>
> I don't quite agree with your list, either, though.  Here's what I
> think, based purely on brute-force with a computer, using as "seeds"
> all the numbers up to 100000:
>
> 1 2 3 4 5 6 7 8 9 11 22 33 44 55 66 77 88 99 110 111 121 132 143 154
> 165 176 187 222 333 444 555 666 777 888 999 1110 1111 1221 1332 1443
> 1554 1665 1776 1887 1998 2109 2220 2222 2331 2442 2553 2664 2775 2886
> 3108 3330 3333 3552 3774 3996 4218 4440 4444 4662 4884 5106 5328 5555
> 6666 7777 8888 9999 11110 11111 12221 13332 14443 15554 16665 17776
> 18887 19998 21109 22220 22222 23331 24442 25553 26664 27775 28886
> 29997
>
> which leaves me wondering, for instance, why I got 29997 but not 2997
> in there.  But now I see it's a nice little bit of combinatorics: 108
> goes to 1998 (six permutations, so effectively each spot is
> (1+0+8+1+0+8) so we get 1800 + 180 + 18), while 1008 goes to 29997
> (twelve permutations, so each spot is (1+0+0+8+1+0+0+8+1+0+0+8) so we
> get 27000 + 2700 + 270 + 27).
>
> So I think my above list of terms are correct.  Differences from your
> list, Diana, are that I have 333 and 1111 and 2222 and 2331 and 2553
> and 2775 and 3333 which are missing from your list.  (2331 for example
> comes from 399 -> 399 + 939 + 993, and any list of identical digits
> maps to itself, so 333 -> 333.)
>
> Diana, would you verify that you agree with my list of terms and then
> if you do please submit the corrected terms for this sequence?
>
> Thanks,
> --Joshua Zucker
>
>
> On 7/11/07, Diana Mecum <diana.mecum at gmail.com> wrote:
> > Sequence Fans,
> >
> > I am looking at sequence A071267.
> >
> > %I A071267
> > %S A071267
> > 2,4,6,8,10,11,12,14,16,18,22,33,44,55,66,77,88,99,101,110,121,132,143,
> > %T A071267 154,165,176,187,198,111,222,666,888,1110
> > %N A071267 Numbers which can be expressed as the sum of all distinct
> digit
> > permutations of
> >                some number k.
> > %C A071267 222 can be expressed so in two different ways i.e. 222= 200
> +020
> > + 002 as well as
> >                222= 101 +110 +011. Question: find a number which can be
> so
> > expressed
> >                 in n different ways.
> > %e A071267 1110 is a member as a sum of all distinct permutations of
> 104.
> > i.e. 104,140,410,
> >                401,014,041.
> > %Y A071267 Sequence in context: A081472 A097660 A067030 this_sequence
> > A072427 A050420 A096922
> > %Y A071267 Adjacent sequences: A071264 A071265 A071266 this_sequence
> A071268
> > A071269 A071270
> > %K A071267 base,more,nonn
> > %O A071267 1,1
> > %A A071267 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 01 2002
> >
> > When I try to follow the rule for generating the sequence numbers, I get
> the
> > following list;
> >
> > 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
> > 121, 132, 143, 154, 165,
> >  176, 187, 222, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443,
> 1554,
> > 1665, 1776, 1887,
> >  1998, 2109, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996
> >
> > Can someone explain why my list differs from the original? I am not
> > understanding the hypothesis to generate the original list.
> >
> > Diana M.
> >
> > --
> > "God made the integers, all else is the work of man."
> > L. Kronecker, Jahresber. DMV 2, S. 19.
>



-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From martin_n_fuller at btinternet.com  Thu Jul 12 15:53:02 2007
From: martin_n_fuller at btinternet.com (Martin Fuller)
Date: Thu, 12 Jul 2007 14:53:02 +0100 (BST)
Subject: 3 seqs that should be the same but are not
In-Reply-To: <001501c7c484$b79a7a30$25933e44@Dell>
Message-ID: <181140.23791.qm@web86606.mail.ukl.yahoo.com>

A074455 and A087300 should be the same: A074455 has the right description and
came first, A087300 has the right values.

I get a different value to OEIS for A074457 as well as for A074455.

See below for the current values and my calculations in PARI.  The PARI code
gives the same value as A087300 and is consistent when increasing the
precision, so I think it's right.

Martin Fuller

PS Can someone explain why the dimension with maximum surface area is exactly 2
more than the dimension with maximum volume?

PPS All these constants are based on geometers dimension which is one more than
topologists dimension (see the comments at MathWorld and compare with the
formulae at PlanetMath
http://planetmath.org/encyclopedia/VolumeOfTheNSphere.html)

OEIS
A087300
5.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926
A074455
5.25694640486057678013283838869076923661901723718321485750987966544135040807732427416016036408330066793834
A074457
7.25694640486057678013283838869076923661901723718321485750987966581295771725141939748631782716397233020121
A074454
5.27776802111340099728214586417284638752999928451017356776163734021486412730547017110062048407258401284645
A074456
33.1611944849620026918630240155829735800472328410872585131001181554037565464718434466607460949351387694776

PARI/GP 
A074455
5.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926957
A074457
7.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926957
A074454
5.27776802111340099728214586417284638752999928451017356776163734021486412730547017110062048407258401284645
A074456
33.1611944849620026918630240155829735800472328410872585131001181554037565464718434466607460949351387694776

/* PARI/GP code */
hyperspheresurface(d)=2*Pi^(d/2)/gamma(d/2)
hyperspherevolume(d)=hyperspheresurface(d)/d

FindMax(fn_x,lo,hi)=
{
  local(oldprecision, x, y, z);
  oldprecision = default(realprecision);
  default(realprecision, oldprecision+10);

  while (hi-lo > 10^-oldprecision,
    while (1,
      z = vector(2, i, lo*(3-i)/3 + hi*i/3);
      y = vector(2, i, eval(Str("x = z[" i "]; " fn_x)));
      if (abs(y[1]-y[2]) > 10^(5-default(realprecision)), break);
      default(realprecision, default(realprecision)+10);
    );
    if (y[1] < y[2], lo = z[1], hi = z[2]);
  );

  default(realprecision, oldprecision);
  (lo + hi) / 2.
}

default(realprecision, 105);
A074455=FindMax("hyperspherevolume(x)", 1, 9)
A074457=FindMax("hyperspheresurface(x)", 1, 9)
A074454=hyperspherevolume(A074455)
A074456=hyperspheresurface(A074457)
/* PARI/GP code ends */

--- "David W. Cantrell" <DWCantrell at sigmaxi.net> wrote:

> ----- Original Message ----- 
> From: "Jonathan Post" <jvospost3 at gmail.com>
> To: <njas at research.att.com>
> Cc: <seqfan at ext.jussieu.fr>; <eric at weisstein.com>; <rgwv at rgwv.com>; 
> "jonathan post" <jvospost2 at yahoo.com>
> Sent: Thursday, July 12, 2007 13:12
> Subject: Re: 3 seqs that should be the same but are not
> 
> 
> > Why should "surface area" (A074457) equal "volume" (A087300)?
> 
> They're not the same constant, but the constant given in A074457 is 
> simply 2 more than the constant given in A087300. Both entries should 
> mention that, I think.
> 
> There are also two errors in the title of A087300. It should read:
> Decimal expansion of value of d for which volume of d-dimensional unit 
> ball is maximized.
> [Note that it is necessary to specify that it's a _unit_ ball.]
> 
> David 
> 
> 





From maximilian.hasler at gmail.com  Fri Jul 13 04:16:00 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Thu, 12 Jul 2007 22:16:00 -0400
Subject: link to 3n+1 ?
Message-ID: <3c3af2330707121916p1f22271bxb9b1a6eac4a04415@mail.gmail.com>

%I A001281
%S A001281 2,1,8,2,14,3,20,4,26,5,32,6,38,7,44,8,50,9,56,10,62,11,68,12,74,13,80,
%N A001281 Image of n under n->n/2 if n even, n->3n-1 if n odd.
%Y A001281 Cf. A037082.

I may be wrong but at first glance, the xref given in this record
refers to something completely unrelated(?) (Primes of the form n!!! +
1),
shouldn't it rather refer to

A006370  Image of n under the `3x+1' map.

or the like? (I could not spot an obvious typo comparing with id's at
http://www.research.att.com/~njas/sequences/Sindx_3.html#3x1

On that token,

%S A008908 1,2,8,3,6,9,17,4,20,7,15,10,10,18,18
%N A008908 Number of halving and tripling steps to reach 1 in `3x+1' problem.
%Y A008908 a(n) = A006577(n) + 1.

seems inconsistent with

%S A006577 0,1,7,2,5,8,16,3,19,6,14,9,9,17,17,4,12,
%N A006577 Number of halving and tripling steps to reach 1 in `3x+1' problem.

(the %N should not be identical if values differ)

M.H.



I've now added all the sequences that were submitted
to this point in time, including those that were "lost".

[The "lost" ones have numbers in the range A130707 - A130742.
There may be some duplicates there - let me know if you notice
any.  I checked, but not very thoroughly.  Getting the "lost" ones
out of the log files was a lot of work.]

The next project is to process "Comments".  There are about 500,

In the meantime, please remember that the OEIS is on vacation.

Neil




From njas at research.att.com  Fri Jul 13 16:12:16 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Fri, 13 Jul 2007 10:12:16 -0400 (EDT)
Subject: OEIS status report
Message-ID: <200707131412.l6DECGEp29694119@fry.research.att.com>

so this will take a while.
Return-Path: <olivier.gerard at gmail.com>
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Message-ID: <335cf8920707131046v25a23917s3adf511e259b60c at mail.gmail.com>
Date: Fri, 13 Jul 2007 19:46:07 +0200
From: "Olivier Gerard" <ogerard at ext.jussieu.fr>
Sender: olivier.gerard at gmail.com
To: seqfan at ext.jussieu.fr
Subject: [seqfan] Online again
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Dear Members,
As the last few messages have proven, the mailing listserver for SeqFan is up again and in good condition.
Have a pleasant day.
Olivier GERARD
====================================
Chers abonn??s de la liste seqfan,
Comme les quelques messages r??cents l'ont montr??,le syst??me informatique de notre liste de discussionest de nouveau op??rationnel.
A tous une bonne journ??e,
Olivier GERARD




From nholmes at leven.comp.utas.edu.au  Sat Jul 14 13:38:18 2007
From: nholmes at leven.comp.utas.edu.au (Neville Holmes)
Date: Sat, 14 Jul 2007 21:38:18 +1000 (EST)
Subject: A008776 and A025192 Duplicates ?
Message-ID: <20070714113818.9533E106B5@leven.comp.utas.edu.au>

A008776 and A025192 look the same to me.  Am I missing something ?
-------------------------------------------------------------------------
Neville Holmes                 E-mail: Neville.Holmes at utas.edu.au
School of Computing            Phone : +61 363 243 393 or (03) 63 243 393
University of Tasmania         Fax   : +61 363 243 368 or (03) 63 243 368
Locked Bag 1-359  Launceston 7250 ---------------------------------------



Neville,  They are not quite the same - one has an 
initial 1.  The relation between them is explicitly mentioned
in the entries.

Neil




From njas at research.att.com  Sat Jul 14 15:28:12 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Sat, 14 Jul 2007 09:28:12 -0400 (EDT)
Subject: A008776 and A025192 Duplicates ?
Message-ID: <200707141328.l6EDSCSj27853097@fry.research.att.com>



A124502 is essentially the same as A108458?


http://www.research.att.com/~njas/sequences/A108458

http://www.research.att.com/~njas/sequences/A124502

V.
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From aomunagi at gmail.com  Mon Jul 16 17:52:58 2007
From: aomunagi at gmail.com (Augustine Munagi)
Date: Mon, 16 Jul 2007 17:52:58 +0200
Subject: Possible duplicate
In-Reply-To: <102c01c7c6d4$a4b992a0$3701f0d5@speedy>
References: <102c01c7c6d4$a4b992a0$3701f0d5@speedy>
Message-ID: <e1c0995c0707160852t74d789b8yce96e6910d7cbc6a@mail.gmail.com>

Hi Seqfans,

They should not be the same unless the definition of A108458 is
modified. A108458 needs improvement.
They are not transposes, even though A108458(n,k)=A124502(n+k,k+1) for
fixed k. By definition A108458 is infinite both downwards and to the
right while A124502 is finite to the right.



On 7/15/07, Vladeta Jovovic <vladeta at eunet.yu> wrote:
>
>
>
> A124502 is essentially the same as A108458?
>
> http://www.research.att.com/~njas/sequences/A108458
>
> http://www.research.att.com/~njas/sequences/A124502
>
> V.




From maxale at gmail.com  Tue Jul 17 06:20:25 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Mon, 16 Jul 2007 21:20:25 -0700
Subject: A091902 is duplicate of A067698
Message-ID: <d3dac270707162120j4d5d510bhe25aa99d85a2a8b2@mail.gmail.com>

SeqFan

A091902 is a duplicate of A067698. They differ only with an extra
first term 1 in A091902, which actually should not be there since
log(log(n)) in not defined for n=1.
A067698 is much developed than A091902, missing only a link to Robin's
Theorem. So, I suggest to add a link to Robin's Theorem to A067698 and
remove A091902 from OEIS.

Max



I have recently submitted these sequences, which have already appeared in 
the database.

>%I A131789
>%S A131789 1,2,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,2,1,1,1,1,1,3,1
>%N A131789 a(n) = the length of the nth run of similar consecutive values 
>in the sequence A000005. (A000005(n) = the number of positive divisors of n.)
>%e A131789 Runs of sequence A000005: (1), (2,2), (3), (2), (4), (2), (4), 
>(3), (4), (2), (6), (2), (4,4), (5), (2), (6), (2), (6), (4,4), (2), (8), 
>(3),...
>%Y A131789 A000005,A131790
>%O A131789 1
>%K A131789 ,more,nonn,


>%I A131790
>%S A131790 1,1,10,1,5,1,3,1,5,1
>%N A131790 a(n) = the length of the nth run of similar consecutive values 
>in the sequence A131789.
>%e A131790 Runs of sequence A131789: (1), (2), (1,1,1,1,1,1,1,1,1,1), (2), 
>(1,1,1,1,1), (2), (1,1,1), (2), (1,1,1,1,1,), (3),...
>%Y A131790 A000005,A131789
>%O A131790 1
>%K A131790 ,more,nonn,


It seems VERY likely to me that there is no infinite string of 1's, or of 
anything else, in sequence A131789 (ie. the terms of A131790 are all 
finite).

Can it be PROVED that all terms of A131790 are finite, possibly using 
Hardy and Wright or some other such reference?

A harder question to answer:
We can define sequence S(m) = {s(m,n)}, where s(m,n) = the length of the 
nth run of similar consecutive values in the sequence S(m-1), where S(0) 
= sequence A000005.
(And S(1) = A131789, S(2) = A131790, of course.)

Is every term of S(m) finite for every m = positive integer?

It seems intuitive obvious that, yes, all terms are finite. But a proof 
would be harder to produce.


Thanks,
Leroy Quet




From qq-quet at mindspring.com  Tue Jul 17 16:45:59 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Tue, 17 Jul 07 08:45:59 -0600
Subject: Lengths Of Runs In The #-Of-Divisors Sequence
Message-ID: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>


Just for fun (?):
http://www.cetteadressecomportecinquantesignes.com/WordsPosition.htm
Best,
?.





From joshua.zucker at gmail.com  Tue Jul 17 21:44:10 2007
From: joshua.zucker at gmail.com (Joshua Zucker)
Date: Tue, 17 Jul 2007 12:44:10 -0700
Subject: Lengths Of Runs In The #-Of-Divisors Sequence
In-Reply-To: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
References: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
Message-ID: <721e81490707171244r38763dedj89e40b5e5d7bee5b@mail.gmail.com>

On 7/17/07, Leroy Quet <qq-quet at mindspring.com> wrote:
> I have recently submitted these sequences, which have already appeared in
> the database.

Should be easy to extend -- is it easier to post here first, asking
someone with a computer to extend them for you, and then submit enough
terms to begin with?

> It seems VERY likely to me that there is no infinite string of 1's, or of
> anything else, in sequence A131789 (ie. the terms of A131790 are all
> finite).
>
> Can it be PROVED that all terms of A131790 are finite, possibly using
> Hardy and Wright or some other such reference?

No need for Hardy and Wright, just Euclid -- there are infinitely many
primes.  Plus there are not any consecutive primes after 2.  Hence the
primes (2s) partition the list into finite chunks.

--Joshua Zucker




From joshua.zucker at gmail.com  Tue Jul 17 21:45:14 2007
From: joshua.zucker at gmail.com (Joshua Zucker)
Date: Tue, 17 Jul 2007 12:45:14 -0700
Subject: Lengths Of Runs In The #-Of-Divisors Sequence
In-Reply-To: <721e81490707171244r38763dedj89e40b5e5d7bee5b@mail.gmail.com>
References: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
	 <721e81490707171244r38763dedj89e40b5e5d7bee5b@mail.gmail.com>
Message-ID: <721e81490707171245r7865a036ude11b441e53b8ae7@mail.gmail.com>

On 7/17/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
> On 7/17/07, Leroy Quet <qq-quet at mindspring.com> wrote:

> > It seems VERY likely to me that there is no infinite string of 1's, or of
> > anything else, in sequence A131789 (ie. the terms of A131790 are all
> > finite).
> >
> > Can it be PROVED that all terms of A131790 are finite, possibly using
> > Hardy and Wright or some other such reference?
>
> No need for Hardy and Wright, just Euclid -- there are infinitely many
> primes.  Plus there are not any consecutive primes after 2.  Hence the
> primes (2s) partition the list into finite chunks.

Wups, I think that's a proof for A131789, not for A131790.

--Joshua



Hello all,

I was looking at A069803 - Smaller of two consecutive palindromic primes:	2, 3, 5, 7, 181, 787, 919 
Conjectured to be complete. 

I am interested in seeing a proof that 919 is actually the largest palindromic prime such that the next prime is palindromic. 
I checked up to 10^8 with Mathematica coding. 

Also, it is obvious that the distance from a palindrome n to the next one is more than Sqrt(n/10). It is clear that prime gaps grow slower than that. Looking at the prime gaps sequence A053303, it is easy to prove that 919 is the last number like that up to 10^16.

Is there a bound for prime gaps that proves that the gaps are less than Sqrt(n/10) starting from some n?

Tanya


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From tanyakh at TanyaKhovanova.com  Wed Jul 18 00:12:20 2007
From: tanyakh at TanyaKhovanova.com (Tanya Khovanova)
Date: Tue, 17 Jul 2007 15:12:20 -0700
Subject: 919 conjecture
In-Reply-To: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
References: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
Message-ID: <200707171512.AA1076494578@TanyaKhovanova.com>

On 7/17/07, Leroy Quet <qq-quet at mindspring.com> wrote:

> Can it be PROVED that all terms of A131790 are finite, possibly using
> Hardy and Wright or some other such reference?

It would follow from the statement:

There exist infinitely many positive integers k such that both 2k+1
and 3k+1 are prime.
I do not have a proof that this statement is true (it may be as hard
as twin prime conjecture) but heuristic arguments ala Hardy--Wright
suggest that it is ture.

Now, if for some k>1 both 2k+1 and 3k+1 are prime then 3(2k+1)=6k+3
and 2(3k+1)=6k+2 both have exactly 4 divisors, implying that
A000005(6k+3)=A000005(6k+2)=4 belong to the same run in A000005 and
the length of this run is greater than 1. In other words, in A131789
elements greater than 1 appear infinitely often.
It is also clear that for each prime p >= 5, A000005(p)=2 forms its
own run of length 1 in A000005, implying that in A131789 elements
equal 1 appear infinitely often.
Therefore, all runs in A131789 are finite and so are elements of A131790.

P.S. btw, the sequence of k such that both 2k+1 and 3k+1 are prime
seems to be missing in OEIS. It starts with:
2, 6, 14, 20, 26, 36, 50, 54, 74, 90, 116, 140, 146, 174, 200, 204,
210, 224, 230, 270, 284, 306, 330, 336, 350, 354, 384, 404, 410, 426,
440, 476, 510, 516, 554, 564, 596, 600, 624, 644, 650, 704, 714, 726,
740, 746, 834, 846, 894, 930, 944, 950

Regards,
Max




From franktaw at netscape.net  Wed Jul 18 05:21:12 2007
From: franktaw at netscape.net (franktaw at netscape.net)
Date: Tue, 17 Jul 2007 23:21:12 -0400
Subject: 919 conjecture
In-Reply-To: <200707171512.AA1076494578@TanyaKhovanova.com>
References: <200707171512.AA1076494578@TanyaKhovanova.com>
Message-ID: <8C99701D3845E00-E70-2A8D@FWM-D43.sysops.aol.com>

No.

It is an open problem of long standing to show that there is always a 
prime between n^2 and (n+1)^2.  This means that we can't prove that 
prime gaps are always less than sqrt(n) for n sufficiently large.

Franklin T. Adams-Watters

-----Original Message-----
From: Tanya Khovanova <tanyakh at TanyaKhovanova.com>

Hello all,

I was looking at A069803 - Smaller of two consecutive palindromic 
primes:   2,
3, 5, 7, 181, 787, 919
Conjectured to be complete.

I am interested in seeing a proof that 919 is actually the largest 
palindromic
prime such that the next prime is palindromic.
I checked up to 10^8 with Mathematica coding.

Also, it is obvious that the distance from a palindrome n to the next 
one is
more than Sqrt(n/10). It is clear that prime gaps grow slower than 
that. Looking
at the prime gaps sequence A053303, it is easy to prove that 919 is the 
last
number like that up to 10^16.

Is there a bound for prime gaps that proves that the gaps are less than
Sqrt(n/10) starting from some n?

Tanya



________________________________________________________________________
Check Out the new free AIM(R) Mail -- Unlimited storage and 
industry-leading spam and email virus protection.




From aomunagi at gmail.com  Wed Jul 18 17:16:54 2007
From: aomunagi at gmail.com (Augustine Munagi)
Date: Wed, 18 Jul 2007 17:16:54 +0200
Subject: Possible duplicate
In-Reply-To: <e1c0995c0707160852t74d789b8yce96e6910d7cbc6a@mail.gmail.com>
References: <102c01c7c6d4$a4b992a0$3701f0d5@speedy>
	 <e1c0995c0707160852t74d789b8yce96e6910d7cbc6a@mail.gmail.com>
Message-ID: <e1c0995c0707180816v27f11abj37566c3304af1ec6@mail.gmail.com>

Hi Seqfan,
Additionally, the relation A108458(n,k)=A124502(n+k,k+1) shows that
the sequences are related like "multichoose" to "choose". I suggest a
merging of the sequences by inserting the definition of A108458 as a
comment in A124502 with the above relation.
Else A108458 as presently displayed is inaccurate, and indeed leaves
us with a duplicate.



On 7/16/07, Augustine Munagi <aomunagi at gmail.com> wrote:
> Hi Seqfans,
>
> They should not be the same unless the definition of A108458 is
> modified. A108458 needs improvement.
> They are not transposes, even though A108458(n,k)=A124502(n+k,k+1) for
> fixed k. By definition A108458 is infinite both downwards and to the
> right while A124502 is finite to the right.
>
>
>
> On 7/15/07, Vladeta Jovovic <vladeta at eunet.yu> wrote:
> >
> >
> >
> > A124502 is essentially the same as A108458?
> >
> > http://www.research.att.com/~njas/sequences/A108458
> >
> > http://www.research.att.com/~njas/sequences/A124502
> >
> > V.
>




From joshua.zucker at gmail.com  Wed Jul 18 23:19:32 2007
From: joshua.zucker at gmail.com (Joshua Zucker)
Date: Wed, 18 Jul 2007 14:19:32 -0700
Subject: Lengths Of Runs In The #-Of-Divisors Sequence
In-Reply-To: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
References: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
Message-ID: <721e81490707181419y19dc932fy51af3d71b8b0cf0@mail.gmail.com>

On 7/17/07, Leroy Quet <qq-quet at mindspring.com> wrote:
> A harder question to answer:
> We can define sequence S(m) = {s(m,n)}, where s(m,n) = the length of the
> nth run of similar consecutive values in the sequence S(m-1), where S(0)
> = sequence A000005.
> (And S(1) = A131789, S(2) = A131790, of course.)
>
> Is every term of S(m) finite for every m = positive integer?
>
> It seems intuitive obvious that, yes, all terms are finite. But a proof
> would be harder to produce.

There seems to be a strongly alternating pattern:
Divisor sequence: 1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6 4 4 2 8 3 4
4 6 2 8 2 6 4 4 4 9 2 4 4 8 2 8 2 6 6 4 2 10 3 6 4 6 2 8 4 8 4 4 2 12
2 4 6 7 4 8 2 6 4 8 2 12 2 4 6 6 4 8 2 10 5 4 2 12 4 4 4 8 2 12 4 6 4
4 4 12 2 6 6 9

Run lengths in divisor sequence: 1 2 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2
1 1 1 2 1 1 1 1 1 3 1 1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 3 1 1 1 1 1 3 1 1 2 1 1 1 1 2
1 1 1 1 1 1 1 1 1 1 2

Run lengths in that: 1 1 10 1 5 1 3 1 5 1 2 1 4 1 11 1 16 1 8 1 5 1 2
1 4 1 10 2 2 1 9 2 4 1 1 2 9 1 11 1 4 1 10 1 10 1 1 1 6 1 1 1 10 1 9 1
7 1 30 1 9 2 1 1 22 1 4 2 8 1 28 1 4 1 4 1 4 1 33 1 3 1 9 1 5 1 26 1
18 1 4 1 5 1 10 1 9 1 3 1

Run lengths in that: 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 3 1 3 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1

And then: 1 25 1 4 1 10 1 1 1 10 1 55 1 15 1 4 1 63 1 3 1 12 1 33 1 5
1 32 1 9 1 7 1 3 1 13 1 56 1 61 1 5 1 103 1 47 1 17 1 13 1 25 1 5 1 5
1 47 1 3 1 21 1 7 1 11 1 1 1 17 1 3 1 1 1 8 1 1 1 5 1 7 1 9 1 2 1 15 1
36 1 5 1 11 1 7 1 1 1 15

And then: 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3
1 1 1 3 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1

And then: 6 1 57 1 3 1 1 1 17 1 63 1 7 1 9 1 101 1 17 1 43 1 13 1 27 1
129 1 37 1 17 1 9 1 39 1 15 1 15 1 45 1 27 1 11 1 3 1 19 1 41 1 5 1 51
1 9 1 47 1 33 1 15 1 35 1 7 1 7 1 15 1 13 1 29 1 53 1 25 1 9 1 23 1 69
1 9 1 5 1 35 1 13 1 9 1 9 1 37 1

And then: 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

And then: 5 1 105 1 3 1 49 1 29 1 5 1 31 1 15 1 9 1 15 1 17 1 5 1 61 1
9 1 1 1 1 1 1 1 13 1 11 1 11 1 41 1 51 1 19 1 73 1 5 1 55 1 31 1 41 1
67 1 133 1 5 1 5 1 85 1 9 1 7 1 61 1 11 1 5 1 1 1 15 1 29 1

And so on, where alternately you get (long strings of 1s with the
occasional other number) and (alternately 1s and other numbers, with
occasionally 2 or 3 1s in a row).

By the way, using the first million terms of A000005, I get only the
first 82 terms of S(8), so it'll take some work to explore this any
further numerically.

I don't think it's at all obvious that this thing won't eventually
yield a sequence of all 1s, even if it is intuitively obvious to Leroy
Quet, it's sure not obvious to me!

--Joshua Zucker
PS: I also submitted extensions of A131789 and A131790 as part of this work.



Joshua Zucker wrote:
>I don't think it's at all obvious that this thing won't eventually
>yield a sequence of all 1s, even if it is intuitively obvious to Leroy
>Quet, it's sure not obvious to me!

I think my use of the phrase "intuitively obvious" is quite an 
exaggeration.
It just seems *likely* to me that the terms of each S(m) are finite. 
Nothing is really obvious.

:)

Thanks,
Leroy Quet




From qq-quet at mindspring.com  Wed Jul 18 23:44:20 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Wed, 18 Jul 07 15:44:20 -0600
Subject: Lengths Of Runs In The #-Of-Divisors Sequence
Message-ID: <E1IBHMR-00036C-00@pop05.mail.atl.earthlink.net>

This is a multi-part message in MIME format.

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Hello SeqFans,
could someone check and compute a hundred terms or so of the seq below?
=20
4,6,9,21,22,25,33,39,46,49,51,54,...
=20
Integers n whose "ordered concatenation" of all divisors (except 1 and n) i=
s a prime.
=20
 n     div.         prime
 4 -> 1,2,4     ->    2
 6 -> 1,2,3,6   ->   23
 9 -> 1,3,9     ->    3
21 -> 1,3,7,21  ->   37
22 -> 1,2,11,22 ->  211
25 -> 1,5,25    ->    5
33 -> 1,3,11,33 ->  311
39 -> 1,3,13,39 ->  313
...
26 is not a member :
=20
26 -> 1,2,13,26 -> 213 -> 1,3,71,213 -> 371 -> 1,7,53,371 -> 753 -> 1,3,251=
,753 -> 3251 is prime
=20
but 753 will be a member.
=20
=20
I like 54:
=20
54 -> 1,2,3,6,9,18,27,54 -> 23691827 prime
=20
Best,
=C9.
=20
-----
=20
(I've used Magma to compute the factorization, plus Edwin Clarkk's advice:
=20
> [Math-Fun]
>
> (...)
> Use for example the command Divisors(12345678); at this site
>
>   http://magma.maths.usyd.edu.au/calc/
>
> and click on evaluate. You will get
>
> [ 1, 2, 3, 6, 9, 18, 47, 94, 141, 282, 423, 846, 14593, 29186, 43779,
> 87558, 131337, 262674, 685871, 1371742, 2057613, 4115226, 6172839,
> 12345678 ]
>
> Total time: 0.350 seconds, Total memory usage: 6.46MB


=20
=20
=20
Please do not send any message during this period as you will
not be able to check whether it has been received or not by
the mailing list robot.  Anyway, they would be distributed
only after the servers are rebooted.

I will send a mail to the list so that everyone knows operations
are back to normal.  Thanks in advance for your patience and
your discipline in that matter.

with my best regards,

Olivier GERARD

PS: As usual, any inquiries about this matter should be directed
to me (olivier.gerard at gmail.com or olivier.gerard at paris7.jussieu.fr)
and not to the list.

=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D

Chers abonn=E9s de la liste seqfan

Arr=EAt temporaire du serveur h=E9bergeant la liste

JEUDI 12 Juillet 18h, heure de Paris, pour les fran=E7ais,

jusqu'au

VENDREDI 13 Juillet au matin (probablement 9h)


S'il vous plait, n'envoyez pas de message pendant cette p=E9riode,
vous ne pourriez pas savoir si il a =E9t=E9 bien re=E7u, et vous risqueriez
de provoquer une confusion ou des doublons.
J'enverrai un courier =E0 la liste quand la situation sera revenue
=E0 la normale.

Merci d'avance pour votre patience et votre discipline.


Cordialement,

Olivier GERARD


PS: Comme d'habitude, si vous avez des questions =E0 ce sujet, =E9crivez
moi directement (olivier.gerard at gmail.com ou
olivier.gerard at paris7.jussieu.fr ) et pas =E0 la liste.





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<BODY>
<DIV id=3DidOWAReplyText34525 dir=3Dltr>
<DIV dir=3Dltr><FONT face=3D"Courier New" color=3D#000000 size=3D2>Hello=20
SeqFans,</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>could someone check and =
compute a=20
hundred terms or so of the seq below?</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New"=20
size=3D2>4,6,9,21,22,25,33,39,46,49,51,54,...</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>Integers n&nbsp;whose "o=
rdered=20
concatenation" of all divisors (except 1 and n) is a prime.</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&nbsp;n&nbsp;&nbsp;&nbsp=
;&nbsp;=20
div.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;prime</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&nbsp;4 -&gt;=20
1,2,4&nbsp;&nbsp;&nbsp;&nbsp; -&gt; &nbsp; &nbsp;2</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&nbsp;6 -&gt; 1,2,3,6&nb=
sp;&nbsp;=20
-&gt;&nbsp;&nbsp; 23</FONT></DIV></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&nbsp;9 -&gt; 1,3,9&nbsp=
;&nbsp;=20
&nbsp; -&gt;&nbsp;&nbsp;&nbsp; 3</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>21 -&gt; 1,3,7,21&nbsp;=
=20
-&gt;&nbsp;&nbsp; 37</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>22 -&gt; 1,2,11,22 -&gt;=
&nbsp;=20
211</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>25 -&gt; 1,5,25&nbsp;&nb=
sp;&nbsp;=20
-&gt;&nbsp;&nbsp;&nbsp; 5</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>33 -&gt; 1,3,11,33 -&gt;=
&nbsp;=20
311</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>39 -&gt; 1,3,13,39 -&gt;=
&nbsp;=20
313</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>...</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>26 is not a member :</FO=
NT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>26 -&gt; 1,2,13,26 -&gt;=
 213 -&gt;=20
1,3,71,213 -&gt; 371 -&gt; 1,7,53,371 -&gt; 753 -&gt; 1,3,251,753 -&gt; 325=
1 is=20
prime</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>but 753 will be a=20
member.</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>I like 54:</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>54 -&gt; 1,2,3,6,9,18,27=
,54 -&gt;=20
23691827 prime</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>Best,</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>=C9.</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>-----</FONT></DIV>
<DIV dir=3Dltr>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>(I've used Magma to comp=
ute the=20
factorization, plus Edwin Clarkk's advice:</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&gt; [Math-Fun]</FONT></=
DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&gt;</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&gt; (...)</FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2>&gt; Use for example the=
 command=20
Divisors(12345678); at this site<BR>&gt;<BR>&gt;&nbsp;&nbsp; <A=20
href=3D"/exchweb/bin/redir.asp?URL=3Dhttp://magma.maths.usyd.edu.au/calc/"=
=20
target=3D_blank>http://magma.maths.usyd.edu.au/calc/</A><BR>&gt;<BR>&gt; an=
d click=20
on evaluate. You will get<BR>&gt;<BR>&gt; [ 1, 2, 3, 6, 9, 18, 47, 94, 141,=
 282,=20
423, 846, 14593, 29186, 43779,<BR>&gt; 87558, 131337, 262674, 685871, 13717=
42,=20
2057613, 4115226, 6172839,<BR>&gt; 12345678 ]<BR>&gt;<BR>&gt; Total time: 0=
.350=20
seconds, Total memory usage: 6.46MB<BR><BR></FONT></DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT face=3D"Courier New" size=3D2></FONT>&nbsp;</DIV>
<DIV dir=3Dltr><FONT><FONT face=3D"Courier New" size=3D2>Please do not send=
 any=20
message during this period as you will<BR>not be able to check whether it h=
as=20
been received or not by<BR>the mailing list robot.&nbsp; Anyway, they would=
 be=20
distributed<BR>only after the servers are rebooted.<BR><BR>I will send a ma=
il to=20
the list so that everyone knows operations<BR>are back to normal.&nbsp; Tha=
nks=20
in advance for your patience and<BR>your discipline in that matter.<BR><BR>=
with=20
my best regards,<BR><BR>Olivier GERARD<BR><BR>PS: As usual, any inquiries a=
bout=20
this matter should be directed<BR>to me (olivier.gerard at gmail.com or=20
olivier.gerard at paris7.jussieu.fr)<BR>and not to the=20
list.<BR><BR>=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D<BR>=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D<BR><BR>Chers=20
abonn=E9s de la liste seqfan<BR><BR>Arr=EAt temporaire du serveur h=E9berge=
ant la=20
liste<BR><BR>JEUDI 12 Juillet 18h, heure de Paris, pour les=20
fran=E7ais,<BR><BR>jusqu'au<BR><BR>VENDREDI 13 Juillet au matin (probableme=
nt=20
9h)<BR><BR><BR>S'il vous plait, n'envoyez pas de message pendant cette=20
p=E9riode,<BR>vous ne pourriez pas savoir si il a =E9t=E9 bien re=E7u, et v=
ous=20
risqueriez<BR>de provoquer une confusion ou des doublons.<BR>J'enverrai un=
=20
courier =E0 la liste quand la situation sera revenue<BR>=E0 la normale.<BR>=
<BR>Merci=20
d'avance pour votre patience et votre=20
discipline.<BR><BR><BR>Cordialement,<BR><BR>Olivier GERARD<BR><BR><BR>PS: C=
omme=20
d'habitude, si vous avez des questions =E0 ce sujet, =E9crivez<BR>moi direc=
tement=20
(olivier.gerard at gmail.com ou<BR>olivier.gerard at paris7.jussieu.fr ) et pas =
=E0 la=20
liste.<BR><BR><BR><BR></FONT></DIV></FONT>

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From Eric.Angelini at kntv.be  Thu Jul 19 22:38:44 2007
From: Eric.Angelini at kntv.be (Eric Angelini)
Date: Thu, 19 Jul 2007 22:38:44 +0200
Subject: Divisors concatenated shape a prime
Message-ID: <F05775148D000C4B9321A5113D53CB7D50D20B@KNTV-SERVER01.kntv.local>

Sorry for the bad header -- and the old "tail"... I'm not working on my  "normal" computer and quite confused :-/
Best,
?.

________________________________


Hello SeqFans,
could someone check and compute a hundred terms or so of the seq below?
 
4,6,9,21,22,25,33,39,46,49,51,54,...
 
Integers n whose "ordered concatenation" of all divisors (except 1 and n) is a prime.
 
 n     div.         prime
 4 -> 1,2,4     ->    2
 6 -> 1,2,3,6   ->   23
 9 -> 1,3,9     ->    3
21 -> 1,3,7,21  ->   37
22 -> 1,2,11,22 ->  211
25 -> 1,5,25    ->    5
33 -> 1,3,11,33 ->  311
39 -> 1,3,13,39 ->  313
...
26 is not a member :
 
26 -> 1,2,13,26 -> 213 -> 1,3,71,213 -> 371 -> 1,7,53,371 -> 753 -> 1,3,251,753 -> 3251 is prime
 
but 753 will be a member.
 
 
I like 54:
 
54 -> 1,2,3,6,9,18,27,54 -> 23691827 prime
 
Best,
?.
 
-----
 
(I've used Magma to compute the factorization, plus Edwin Clarkk's advice:
 
> [Math-Fun]
>
> (...)
> Use for example the command Divisors(12345678); at this site
>
>   http://magma.maths.usyd.edu.au/calc/
>
> and click on evaluate. You will get
>
> [ 1, 2, 3, 6, 9, 18, 47, 94, 141, 282, 423, 846, 14593, 29186, 43779,
> 87558, 131337, 262674, 685871, 1371742, 2057613, 4115226, 6172839,
> 12345678 ]
>
> Total time: 0.350 seconds, Total memory usage: 6.46MB


 
 
-------------- next part --------------
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From simon.plouffe at gmail.com  Thu Jul 19 22:54:57 2007
From: simon.plouffe at gmail.com (Simon Plouffe)
Date: Thu, 19 Jul 2007 16:54:57 -0400
Subject: Divisors concatenated shape a prime
In-Reply-To: <F05775148D000C4B9321A5113D53CB7D50D20B@KNTV-SERVER01.kntv.local>
References: <F05775148D000C4B9321A5113D53CB7D50D20B@KNTV-SERVER01.kntv.local>
Message-ID: <33a322bc0707191354h70440f5amb6693976ff15c9a5@mail.gmail.com>

hello I made
this simple maple routine :
###############################
with(numtheory):

for k from 5 to 1e99 do:
v0:=divisors(k):

nn:=nops(v0):
if nn > 3 then
v1:=[seq(v0[j],j=2..nn-1)]:
v2:=cat(seq(convert(v1[n],string),n=1..nops(v1))):
v3:=parse(v2):
if isprime(v3) = true then lprint(k,v3) fi:
fi:
od:
#################################

and here is some output from it
(see the attachement)
#################################

Simon plouffe
-------------- next part --------------
6, 23
21, 37
22, 211
33, 311
39, 313
46, 223
51, 317
54, 23691827
58, 229
78, 236132639
82, 241
93, 331
99, 391133
111, 337
115, 523
133, 719
141, 347
142, 271
147, 372149
153, 391751
154, 2711142277
159, 353
162, 236918275481
166, 283
174, 236295887
177, 359
186, 236316293
187, 1117
189, 379212763
201, 367
205, 541
219, 373
226, 2113
235, 547
237, 379
247, 1319
249, 383
253, 1123
262, 2131
267, 389
274, 2137
286, 211132226143
291, 397
294, 23671421424998147
301, 743
318, 23653106159
319, 1129
327, 3109
333, 3937111
355, 571
358, 2179
366, 23661122183
387, 3943129
391, 1723
402, 23667134201
411, 3137
427, 761
459, 39172751153
478, 2239
489, 3163
501, 3167
502, 2251
505, 5101
511, 773
531, 3959177
535, 5107
538, 2269
543, 3181
562, 2281
565, 5113
573, 3191
583, 1153
586, 2293
589, 1931
598, 213232646299
606, 236101202303
622, 2311
639, 3971213
654, 236109218327
657, 3973219
658, 27144794329
679, 797
687, 3229
694, 2347
697, 1741
711, 3979237
721, 7103
726, 23611223366121242363
747, 3983249
753, 3251
759, 311233369253
763, 7109
766, 2383
771, 3257
778, 2389
781, 1171
793, 1361
799, 1747
801, 3989267
813, 3271
822, 236137274411
835, 5167
871, 1367
889, 7127
895, 5179
901, 1753
921, 3307
931, 71949133
934, 2467
939, 3313
943, 2341
949, 1373
957, 311293387319
985, 5197
993, 3331
994, 271471142497
1003, 1759
1006, 2503
1017, 39113339
1041, 3347
1042, 2521
1057, 7151
1077, 3359
1078, 27111422497798154539
1081, 2347
1083, 31957361
1114, 2557
1119, 3373
1135, 5227
1143, 39127381
1147, 3137
1165, 5233
1167, 3389
1186, 2593
1234, 2617
1243, 11113
1254, 23611192233385766114209418627
1294, 2647
1299, 3433
1318, 2659
1339, 13103
1341, 39149447
1347, 3449
1351, 7193
1354, 2677
1362, 236227454681
1366, 2683
1371, 3457
1383, 3461
1387, 1973
1389, 3463
1401, 3467
1405, 5281
1411, 1783
1417, 13109
1426, 223314662713
1435, 573541205287
1438, 2719
1441, 11131
1467, 39163489
1473, 3491
1474, 2112267134737
1477, 7211
1497, 3499
1501, 1979
1513, 1789
1518, 23611222333466669138253506759
1537, 2953
1551, 3113347141517
1581, 317315193527
1594, 2797
1603, 7229
1611, 39179537
1615, 517198595323
1623, 3541
1629, 39181543
1633, 2371
1639, 11149
1641, 3547
1645, 573547235329
1651, 13127
1666, 271417344998119238833
1671, 3557
1686, 236281562843
1713, 3571
1714, 2857
1719, 39191573
1735, 5347
1749, 3113353159583
1774, 2887
1779, 3593
1798, 229315862899
1819, 17107
1821, 3607
1839, 3613
1843, 1997
1851, 3617
1855, 573553265371
1893, 3631
1899, 39211633
1902, 236317634951
1903, 11173
1906, 2953
1909, 2383
1929, 3643
1942, 2971
1963, 13151
1977, 3659
1981, 7283
2001, 323296987667
2013, 3113361183671
2019, 3673
2026, 21013
2031, 3677
2034, 2369181132263396781017
2038, 21019
2043, 39227681
2047, 2389
2059, 2971
2062, 21031
2073, 3691
2077, 3167
2094, 2363496981047
2095, 5419
2103, 3701
2119, 13163
2122, 21061
2127, 3709
2149, 7307
2155, 5431
2157, 3719
2167, 11197
2169, 39241723
2173, 4153
2181, 3727
2199, 3733
2214, 236918274154821232463697381107
2215, 5443
2217, 3739
2245, 5449
2257, 3761
2259, 39251753
2283, 3761
2307, 3769
2317, 7331
2326, 21163
2329, 17137
2343, 3113371213781
2374, 21187
2386, 21193
2391, 3797
2395, 5479
2419, 4159
2421, 39269807
2442, 2361122333766741112224078141221
2443, 7349
2463, 3821
2469, 3823
2479, 3767
2514, 2364198381257
2515, 5503
2527, 719133361
2529, 39281843
2554, 21277
2559, 3853
2566, 21283
2586, 2364318621293
2589, 3863
2605, 5521
2629, 11239
2631, 3877
2637, 39293879
2638, 21319
2641, 19139
2643, 3881
2679, 3194757141893
2721, 3907
2733, 3911
2742, 2364579141371
2757, 3919
2761, 11251
2763, 39307921
2773, 4759
2785, 5557
2787, 3929
2815, 5563
2827, 11257
2839, 17167
2841, 3947
2845, 5569
2866, 21433
2883, 33193961
2901, 3967
2914, 2314762941457
2922, 2364879741461
2923, 3779
2929, 29101
2962, 21481
2967, 3234369129989
2974, 21487
2977, 13229
2983, 19157
2986, 21493
2994, 2364999981497
2998, 21499
3013, 23131
3031, 7433
3039, 31013
3046, 21523
3057, 31019
3097, 19163
3099, 31033
3115, 573589445623
3117, 31039
3118, 21559
3123, 393471041
3133, 13241
3139, 4373
3141, 393491047
3153, 31051
3154, 21938831661577
3178, 27142274541589
3189, 31063
3199, 7457
3202, 21601
3205, 5641
3207, 31069
3226, 21613
3235, 5647
3246, 23654110821623
3247, 17191
3265, 5653
3273, 31091
3286, 23153621061643
3295, 5659
3322, 211221513021661
3333, 311331013031111
3346, 27142394781673
3357, 393731119
3369, 31123
3409, 7487
3415, 5683
3417, 31751672011139
3421, 11311
3439, 19181
3451, 71729119203493
3453, 31151
3459, 31153
3487, 11317
3493, 7499
3505, 5701
3526, 2414382861763
3543, 31181
3573, 393971191
3574, 21787
3579, 31193
3589, 3797
3597, 311331093271199
3609, 394011203
3657, 32353691591219
3661, 7523
3669, 31223
3693, 31231
3711, 31237
3715, 5743
3742, 21871
3747, 31249
3777, 31259
3786, 23663112621893
3787, 7541
3789, 394211263
3799, 29131
3829, 7547
3831, 31277
3841, 23167
3883, 11353
3901, 4783
3921, 31307
3957, 31319
3963, 31321
3973, 29137
3979, 23173
3981, 31327
3987, 394431329
3994, 21997
4006, 22003
4009, 19211
4011, 37211915731337
4039, 7577
4047, 31957712131349
4054, 22027
4063, 17239
4069, 13313
4071, 32359691771357
4074, 23671421429719429158267913582037
4078, 22039
4105, 5821
4126, 22063
4135, 5827
4147, 111329143319377
4165, 5717354985119245595833
4171, 4397
4183, 4789
4195, 5839
4213, 11383
4222, 22111
4249, 7607
4258, 22129
4285, 5857
4303, 13331
4306, 22153
4309, 31139
4342, 213261673342171
4354, 27143116222177
4369, 17257
4381, 13337
4398, 23673314662199
4405, 5881
4429, 43103
4443, 31481
4453, 6173
4458, 23674314862229
4467, 31489
4494, 236714214210721432164274914982247
4501, 7643
4531, 23197
4533, 31511
4546, 22273
4585, 5735131655917
4593, 31531
4623, 32367692011541
4629, 31543
4633, 41113
4653, 39113347991414235171551
4681, 31151
4701, 31567
4711, 7673
4722, 23678715742361
4726, 217341392782363
4749, 31583
4753, 74997679
4762, 22381
4765, 5953
4771, 13367
4803, 31601
4821, 31607
4837, 7691
4843, 29167
4858, 27143476942429
4873, 11443
4881, 31627
4882, 22441
4894, 22447
4923, 395471641
4929, 33153931591643
4941, 392761811835491647
4971, 31657
4981, 17293
4989, 31663
5001, 31667
5002, 24161821222501
5017, 29173
5034, 23683916782517
5062, 22531
5086, 22543
5089, 7727
5097, 31699
5098, 22549
5103, 379212763811892435677291701
5133, 32959871771711
5137, 11467
5143, 37139
5146, 23162831662573
5155, 51031
5161, 13397
5163, 31721
5169, 31723
5191, 29179
5193, 395771731
5206, 219381372742603
5221, 23227
5223, 31741
5242, 22621
5293, 6779
5299, 7757
5305, 51061
5353, 53101
5358, 236193847579411414128289317862679
5371, 41131
5389, 17317
5398, 22699
5401, 11491
5473, 13421
5482, 22741
5511, 311331675011837
5514, 23691918382757
5518, 23162891782759
5533, 11503
5539, 29191
5541, 31847
5542, 217341633262771
5545, 51109
5554, 22777
5593, 71747119329799
5607, 3792163892676238011869
5611, 31181
5619, 31873
5721, 31907
5722, 22861
5746, 2131726341692213384422873
5751, 392771812136391917
5754, 236714214213727441182295919182877
5755, 51151
5761, 7823
5773, 23251
5803, 7829
5823, 396471941
5898, 23698319662949
5917, 6197
5919, 31973
5926, 22963
5941, 13457
5965, 51193
5971, 7853
5983, 31193
5989, 53113
6009, 32003
6019, 13463
6022, 23011
6058, 213262334663029
6081, 32027
6082, 23041
6085, 51217
6087, 32029
6099, 319571073212033
6102, 2369182754113226339678101720343051
6103, 17359
6109, 41149
6139, 7877
6145, 51229
6181, 7883
6187, 23269
6189, 32063
6201, 391339531171594776892067
6207, 32069
6223, 749127889
6226, 211222835663113
6249, 32083
6267, 32089
6286, 27144498983143
6297, 32099
6306, 236105121023153
6309, 397012103
6313, 59107
6331, 13487
6334, 23167
6349, 7907
6354, 236918353706105921183177
6391, 7117783581913
6394, 223461392783197
6406, 23203
6415, 51283
6418, 23209
6423, 32141
6429, 32143
6433, 7919
6439, 47137
6457, 11587
6487, 13499
6493, 43151
6502, 23251
6511, 17383
6522, 236108721743261
6523, 11593
6535, 51307
6538, 27144679343269
6543, 397272181
6558, 236109321863279
6559, 7937
6565, 513651015051313
6601, 72341161287943
6609, 32203
6613, 17389
6639, 32213
6667, 59113
6697, 37181
6706, 27144799583353
6711, 32237
6721, 111347143517611
6727, 731217961
6739, 23293
6742, 23371
6753, 32251
6774, 236112922583387
6787, 11617
6799, 13523
6805, 51361
6811, 749139973
6817, 17401
6862, 24773941463431
6891, 32297
6895, 57351979851379
6901, 67103
6903, 391339591171775317672301
6913, 31223
6921, 397692307
6927, 32309
6978, 236116323263489
7003, 47149
7006, 231621132263503
7023, 32341
7078, 23539
7087, 19373
7089, 317511394172363
7113, 32371
7114, 23557
7123, 17419
7131, 32377
7135, 51427
7143, 32381
7153, 23311
7162, 23581
7186, 23593
7195, 51439
7201, 19379
7233, 32411
7246, 23623
7269, 32423
7273, 71039
7279, 29251
7323, 32441
7327, 17431
7339, 41179
7342, 23671
7354, 23677
7363, 37199
7387, 8389
7401, 32467
7405, 51481
7413, 372135310592471
7435, 51487
7438, 23719
7447, 11677
7467, 319571313932489
7483, 71069
7501, 13577
7509, 32503
7522, 23761
7531, 17443
7534, 23767
7542, 236918419838125725143771
7555, 51511
7593, 32531
7614, 2369182747548194141162282423846126925383807
7633, 17449
7666, 23833
7693, 7491571099
7711, 11701
7713, 398572571
7731, 398592577
7737, 32579
7777, 711771017071111
7801, 29269
7822, 23911
7827, 32609
7834, 23917
7858, 23929
7863, 32621
7881, 337711112132627
7893, 398772631
7897, 53149
7899, 32633
7903, 71129
7909, 11719
7929, 398812643
7939, 17467
7941, 32647
7947, 398832649
7969, 13613
7974, 236918443886132926583987
7986, 23611223366121242363726133126623993
7999, 19421
8002, 24001
8014, 24007
8035, 51607
8038, 24019
8047, 13619
8061, 32687
8065, 51613
8071, 71153
8074, 211223677344037
8079, 32693
8086, 213263116224043
8091, 39293187932612798992697
8098, 24049
8121, 32707
8137, 79103
8139, 32713
8157, 32719
8173, 11743
8182, 24091
8185, 51637
8197, 71171
8227, 19433
8247, 32749
8251, 37223
8266, 24133
8323, 729412032871189
8347, 17491
8365, 573523911951673
8367, 32789
8383, 83101
8391, 32797
8401, 31271
8403, 32801
8409, 32803
8421, 372140112032807
8458, 24229
8494, 231621372744247
8499, 32833
8502, 23613263978109218327654141728344251
8514, 23691118223343668699129198258387473774946141928384257
8529, 32843
8587, 31277
8605, 51721
8634, 236143928784317
8653, 17509
8659, 71237
8661, 32887
8674, 24337
8683, 19457
8727, 32909
8743, 71249
8746, 24373
8751, 32917
8754, 236145929184377
8782, 24391
8791, 59149
8797, 19463
8815, 541432052151763
8817, 32939
8826, 236147129424413
8842, 24421
8847, 399832949
8871, 32957
8874, 236917182934515887102153174261306493522986147929584437
8907, 32969
8913, 32971
8935, 51787
8962, 24481
8983, 13691
8989, 89101
8997, 32999
9019, 29311
9031, 11821
9034, 24517
9061, 131741221533697
9069, 33023
9082, 219382394784541
9094, 24547
9097, 11827
9111, 33037
9117, 3910133039
9147, 33049
9169, 53173
9211, 61151
9217, 13709
9229, 11839
9238, 231621492984619
9246, 23623466769134138201402154130824623
9249, 33083
9271, 73127
9289, 71327
9303, 372144313293101
9306, 236911182233476694991411982824235178461034155131024653
9307, 41227
9313, 67139
9322, 259791181584661
9339, 311332838493113
9355, 51871
9357, 33119
9382, 24691
9393, 331931013033131
9426, 236157131424713
9466, 24733
9493, 11863
9499, 723591614131357
9517, 31307
9523, 89107
9535, 51907
9543, 33181
9549, 3910613183
9553, 41233
9565, 51913
9573, 33191
9586, 24793
9589, 43223
9598, 24799
9609, 33203
9633, 3131939571692475077413211
9658, 211224398784829
9673, 17569
9693, 392735910773231
9745, 51949
9754, 24877
9757, 11887
9763, 13751
9778, 24889
9793, 71399
9798, 23623466971138142213426163332664899
9819, 3910913273
9838, 24919
9841, 13757
9853, 59167
9865, 51973
9886, 24943
9903, 33301
9913, 23431
9934, 24967
9987, 33329
9991, 97103
9993, 33331
10003, 71429
10029, 33343
10041, 33347
10057, 89113
10063, 29347
10077, 33359
10087, 711771319171441
10135, 52027
10171, 71453
10173, 33391
10174, 25087
10179, 39132729398711726135137778311313393
10183, 17599
10203, 319571795373401
10209, 341831232493403
10239, 33413
10249, 37277
10279, 19541
10294, 25147
10297, 71471
10306, 25153
10311, 372149114733437
10326, 236172134425163
10342, 25171
10345, 52069
10351, 11941
10354, 231621673345177
10359, 3911513453
10371, 33457
10378, 25189
10381, 71483
10383, 33461
10405, 52081
10407, 33469
10422, 23691827541933865791158173734745211
10441, 53197
10447, 31337
10461, 311333179513487
10467, 3911633489
10474, 25237
10483, 11953
10519, 67157
10522, 25261
10537, 41257
10539, 3911713513
10587, 33529
10599, 33533
10606, 25303
10615, 511551939652123
10618, 25309
10641, 33547
10681, 11971
10722, 236178735745361
10726, 231621733465363
10743, 33581
10765, 52153
10777, 13829
10783, 41263
10797, 359611771833599
10807, 101107
10809, 3912013603
10827, 392740112033609
10839, 33613
10843, 71549
10849, 19571
10851, 33617
10869, 33623
10902, 23623466979138158237474181736345451
10906, 27141938418213326628757477915585453
10911, 33637
10914, 236173451102107214321642181936385457
10917, 3912133639
10923, 311333319933641
10927, 7492231561
10942, 25471
10947, 341891232673649
10963, 19577
10971, 3923536915920747712193657
10981, 79139
11007, 3912233669
11029, 41269
11061, 3912293687
11107, 29383
11133, 3912373711
11134, 219382935865567
11163, 3611833721
11166, 236186137225583
11167, 13859
11169, 3917517315321965712413723
11179, 71597
11185, 52237
11203, 17659
11217, 33739
11262, 236187737545631
11278, 25639
11281, 29389
11301, 33767
11307, 33769
11313, 392741912573771
11314, 25657
11326, 271480916185663
11335, 52267
11341, 111031
11358, 2369186311262189337865679
11359, 37307
11371, 83137
11386, 25693
11391, 33797
11401, 13877
11406, 236190138025703
11413, 101113
11434, 25717
11473, 7117714910431639
11479, 13883
11482, 25741
11506, 2112252310465753
11511, 3912793837
11517, 3113334910473839
11521, 41281
11539, 111049
11553, 33851
11581, 37313
11589, 33863
11602, 25801
11611, 17683
11623, 59197
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148431, 349477
148453, 532801
148474, 2611221217243474237
148497, 349499
148563, 391751153971291387391650749521
148587, 349529
148599, 3911193357799917120923762771186915011881260745037821135091651149533
148615, 529723
148683, 329871709512749561
148686, 236247814956274343
148702, 214929849999874351
148705, 529741
148714, 274357
148722, 2367142142354170821062321246247874957474361
148761, 391652949587
148801, 178753
148803, 319325757977149601
148809, 349603
148849, 473167
148863, 311133339143347429104138174511114511353349621
148882, 274441
148903, 171932346178378759
148906, 274453
148942, 274471
148947, 3131379393113749649
148978, 274489
148981, 721283
148989, 349663
149001, 349667
149041, 1031447
149091, 349697
149095, 529819
149107, 7174911917983312533043877121301
149163, 372171032130949721
149167, 433469
149203, 314813
149218, 274609
149233, 721319
149329, 592531
149353, 233641
149361, 349787
149386, 2113226661132274693
149403, 349801
149437, 295153
149449, 199751
149493, 349831
149527, 741287521364721361
149539, 1311503
149562, 23679141821426312611872374356171228309106831661821366249274985474781
149613, 349871
149635, 529927
149653, 721379
149662, 274831
149686, 274843
149719, 178807
149722, 274861
149739, 3193757711112137031349210926274047788149913
149757, 349919
149779, 721397
149781, 349927
149785, 5291451033516529957
149817, 349939
149901, 329871723516949967
149902, 224131148262274951
149923, 178819
149946, 2366713420137340274611192238249914998274973
149949, 391666149983
149977, 473191
150099, 350033
150121, 2361107140324616527
150127, 178831
150166, 275083
150238, 2112268291365875119
150253, 971549
150261, 350087
150265, 541205733366530053
150279, 350093
150291, 391669950097
150313, 831811
150321, 389267563168950107
150322, 275161
150333, 350111
150334, 275167
150367, 721481
150421, 359419
150438, 236250735014675219
150454, 275227
150463, 379397
150477, 350159
150493, 721499
150529, 1091381
150538, 275269
150541, 473203
150561, 391672950187
Interrupted

From simon.plouffe at gmail.com  Thu Jul 19 23:10:11 2007
From: simon.plouffe at gmail.com (Simon Plouffe)
Date: Thu, 19 Jul 2007 17:10:11 -0400
Subject: Divisors concatenated shape a prime
In-Reply-To: <F05775148D000C4B9321A5113D53CB7D50D20B@KNTV-SERVER01.kntv.local>
References: <F05775148D000C4B9321A5113D53CB7D50D20B@KNTV-SERVER01.kntv.local>
Message-ID: <33a322bc0707191410x7c82df34qb3b68408bbab7077@mail.gmail.com>

Hello,

And for the number 339066 the prime generated has 255 digits:

here it is :
339066, 2367913141821232627394246546369788191117126138161162182189207234273299322351378414483546567598621702819897966105
311341242144916381794186320932106245726912898372641864347491453826279737180738694125581304114742161461883724219260823767
44843856511113022169533

je continue les calculs.

simon plouffe




From Eric.Angelini at kntv.be  Thu Jul 19 23:19:48 2007
From: Eric.Angelini at kntv.be (Eric Angelini)
Date: Thu, 19 Jul 2007 23:19:48 +0200
Subject: =?iso-8859-1?Q?RE=A0=3A_Divisors_concatenated_shape_a_prime?=
Message-ID: <F05775148D000C4B9321A5113D53CB7D50D20C@KNTV-SERVER01.kntv.local>

 
Geee, Simon !
Simple routine but... great result ! Waow !
Thanks to you and everyone -- this list is a must ;-)
Best,
?.
 
 

________________________________

De: Simon Plouffe [mailto:simon.plouffe at gmail.com]
Date: jeu. 19/07/2007 22:54
?: Eric Angelini
Cc: seqfan at ext.jussieu.fr
Objet : Re: Divisors concatenated shape a prime



hello I made
this simple maple routine :
###############################
with(numtheory):

for k from 5 to 1e99 do:
v0:=divisors(k):

nn:=nops(v0):
if nn > 3 then
v1:=[seq(v0[j],j=2..nn-1)]:
v2:=cat(seq(convert(v1[n],string),n=1..nops(v1))):
v3:=parse(v2):
if isprime(v3) = true then lprint(k,v3) fi:
fi:
od:
#################################

and here is some output from it
(see the attachement)
#################################

Simon plouffe


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From njas at research.att.com  Fri Jul 20 00:27:53 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Thu, 19 Jul 2007 18:27:53 -0400 (EDT)
Subject: Divisors concatenated shape a prime
Message-ID: <200707192227.l6JMRr4e351446@fry.research.att.com>

sequence A037274, of course?  Very similar to yours,
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Message-ID: <d3dac270707191613u57980344yef08bacd01a44fa3 at mail.gmail.com>
Date: Thu, 19 Jul 2007 16:13:08 -0700
From: "Max Alekseyev" <maxale at gmail.com>
To: njas at research.att.com
Subject: Re: Divisors concatenated shape a prime
Cc: seqfan at ext.jussieu.fr, "Eric Angelini" <Eric.Angelini at kntv.be>
In-Reply-To: <200707192227.l6JMRr4e351446 at fry.research.att.com>
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Neil,

It seems that A037274 (and related sequences, e.g., A037275) is
missing the "base" keyword.

Regards,
Max

On 7/19/07, N. J. A. Sloane <njas at research.att.com> wrote:
> Eric,  you know about the "home primes"
> sequence A037274, of course?  Very similar to yours,
> and known to be hard!
> Neil
>




From warut822 at gmail.com  Fri Jul 20 04:32:50 2007
From: warut822 at gmail.com (Warut Roonguthai)
Date: Fri, 20 Jul 2007 09:32:50 +0700
Subject: Lengths Of Runs In The #-Of-Divisors Sequence
In-Reply-To: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
References: <E1IAoM3-0004i7-00@pop05.mail.atl.earthlink.net>
Message-ID: <482644420707191932s56b4979p6fc0c36fd3576d31@mail.gmail.com>

On 7/17/07, Leroy Quet <qq-quet at mindspring.com> wrote:
>
> It seems VERY likely to me that there is no infinite string of 1's, or of
> anything else, in sequence A131789 (ie. the terms of A131790 are all
> finite).
>
> Can it be PROVED that all terms of A131790 are finite, possibly using
> Hardy and Wright or some other such reference?

Since there are infinitely many primes, 1 will appear infinitely often
in A131789. But there cannot be an infinite string of 1's in A131789
because there exist infinitely many m such that d(m) = d(m+1) as
proved by Roger Heath-Brown in 1984. So all terms of A131790 are
finite.

Warut



Simon Plouffe:

> And for the number 339066 the prime generated has 255 digits

Looking at just the record values, from here on...

{record-index, seed-number, digits-in-target-prime}

{41,339066,255}
{42,594594,317}
{43,902538,328}
{44,1750014,341}
{45,2254098,346}
{46,3174138,352}
{47,3467646,354}
{48,3818178,446}
{49,3913434,447}
{50,8795358,501}
{51,9489018,502}
{52,9522414,503}
{53,13891878,511}
{54,14139762,514}
{55,14167494,515}
{56,19803966,522}
{57,23978262,529}
{58,24478146,594}
{59,28289898,673}
{60,35837802,1025}




I'm adding Eric's two new sequences: they will
be A120712 and A120713.  I used Simon Plouffe's 
nice Maple program to generate them, but modified
it to give exactly the same terms as Eric had.

To get an analogue of the home primes, A037274, I suppose
we could iterate the map (k -> concatenation of proper
divisors of k) until we reach a prime; then a(n) would
be the prime we eventually reach when starting with n,
or -1 if we never reach a prime.  But what about a(p)
where p is prime?  I guess we set that to equal p.

And we can set a(1) = 1.  So the sequence starts
1 2 3 2 5 23 7 ...
What is a(8)?
I don't know, but I will put it in the OEIS as A120716.
Hopefully someone will extend it.

Neil




Seqfans,  for the new sequence A120716 that I mentioned,
the analogue of home primes, the big question is,
what is a(8)?
we have 

and the divisors of that last number aree

1, 2, 3, 6, 37, 74, 111, 113, 222, 226, 339, 678, 4181, 8362, 12543, 25086,
3192525397, 6385050794, 9577576191, 19155152382, 118123439689, 236246879378,
354370319067, 360755369861, 708740638134, 721510739722, 1082266109583,
2164532219166, 13347948684857, 26695897369714, 40043846054571, 80087692109142,
30132785470246166539693, 60265570940492333079386, 90398356410738499619079,
180796712821476999238158, 1114913062399108161968641, 2229826124798216323937282,
3344739187197324485905923, 3405004758137816818985309, 6689478374394648971811846,
6810009516275633637970618, 10215014274413450456955927,
20430028548826900913911854, 125985176051099222302456433,
251970352102198444604912866, 377955528153297666907369299,
755911056306595333814738598, 96199682896113474519861511083121,
192399365792226949039723022166242, 288599048688340423559584533249363,
577198097376680847119169066498726, 3559388267156198557234875910075477,
7118776534312397114469751820150954, 10678164801468595671704627730226431,
10870564167260822620744350752392673, 21356329602937191343409255460452862,
21741128334521645241488701504785346, 32611692501782467862233052257178019,
65223385003564935724466104514356038, 402210874188650436967540977838528901,
804421748377300873935081955677057802, 1206632622565951310902622933515586703,
2413265245131902621805245867031173406 ]

If we throw away the first and last terms and concatenate the rest,
is that a prime?  Does someone have access to a really good
prime tester?!

Neil

PS I did not check these calculations and it is 05:00




From pxp at rogers.com  Fri Jul 20 06:17:12 2007
From: pxp at rogers.com (Hans Havermann)
Date: Fri, 20 Jul 2007 00:17:12 -0400
Subject: Divisors concatenated shape a prime
In-Reply-To: <33a322bc0707191410x7c82df34qb3b68408bbab7077@mail.gmail.com>
References: <F05775148D000C4B9321A5113D53CB7D50D20B@KNTV-SERVER01.kntv.local> <33a322bc0707191410x7c82df34qb3b68408bbab7077@mail.gmail.com>
Message-ID: <EA720807-1D72-4D3C-ABA3-05D61696B393@rogers.com>

On 7/20/07, N. J. A. Sloane <njas at research.att.com> wrote:

> If we throw away the first and last terms and concatenate the rest,
> is that a prime?

No, it is not prime.
And it is quite a large number that may be hard to factor. I will let
the PARI/GP factor() function run overnight...

Max




From simon.plouffe at gmail.com  Fri Jul 20 13:43:16 2007
From: simon.plouffe at gmail.com (Simon Plouffe)
Date: Fri, 20 Jul 2007 07:43:16 -0400
Subject: ps Re Divisors concatenated shape a prime
In-Reply-To: <d3dac270707200436x7996f230qdeda7d1a243a730e@mail.gmail.com>
References: <200707201002.l6KA2ZKE432256@fry.research.att.com>
	 <d3dac270707200436x7996f230qdeda7d1a243a730e@mail.gmail.com>
Message-ID: <33a322bc0707200443l833a591gc4abfcff9e172a43@mail.gmail.com>

The number is
23637741111132222263396784181836212543250863192525397638\
    5050794957757619119155152382118123439689236246879378\
    3543703190673607553698617087406381347215107397221082\
    2661095832164532219166133479486848572669589736971440\
    0438460545718008769210914230132785470246166539693602\
    6557094049233307938690398356410738499619079180796712\
    8214769992381581114913062399108161968641222982612479\
    8216323937282334473918719732448590592334050047581378\
    1681898530966894783743946489718118466810009516275633\
    6379706181021501427441345045695592720430028548826900\
    9139118541259851760510992223024564332519703521021984\
    4460491286637795552815329766690736929975591105630659\
    5333814738598961996828961134745198615110831211923993\
    6579222694903972302216624228859904868834042355958453\
    3249363577198097376680847119169066498726355938826715\
    6198557234875910075477711877653431239711446975182015\
    0954106781648014685956717046277302264311087056416726\
    0822620744350752392673213563296029371913434092554604\
    5286221741128334521645241488701504785346326116925017\
    8246786223305225717801965223385003564935724466104514\
    3560384022108741886504369675409778385289018044217483\
    7730087393508195567705780212066326225659513109026229\
    33515586703

and divisible by 13 and 47 if I am not mistaking.

the rest (%/13/47) is not prime either but I can't determine
its factors.

simon plouffe




From jvospost3 at gmail.com  Sat Jul 21 00:35:58 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Fri, 20 Jul 2007 15:35:58 -0700
Subject: Concatenated anti-divisors
Message-ID: <5542af940707201535q1a3f22f5xfd0c2bdb892fab6d@mail.gmail.com>

Non-divisor: a number k which does not divide a given
number x. Anti-divisor: a non-divisor k of x with the
property that k is an odd divisor of 2x-1 or 2x+1, or
an even divisor of 2x.

There are no anti-divisors of 1 and 2.

See A066272 for an equivalent definition and also the
number of terms in each row.

Now, if we concatenate antidivisors of integers n>2, we have an
anti-divisor analogue of A106708.

This should be easy for someone to Mathematica-ize and extend, and
leads to the same kind of fun questions as its origin does.

===========
n  a(n)  factorization
3  2     prime
4  3     prime
5  23    prime
6  4     2^2
7  235   5 * 47
8  35    5 * 7
9  26    2 * 13
10 347   prime
11 237   3 * 79
12 58    2 * 29
13 2359  7 * 337
14 349   prime
15 2610  2 * 3^2 * 5 * 29
16 311   prime
17 235711  7 * 151 * 223
18 45712 2^4 * 2857
19 2313  3^2 * 257
20 3813  3 * 31 * 41
===========

--- The On-Line Encyclopedia of Integer Sequences
<oeis at research.att.com> wrote:

 The following is a copy of the email message that
 was sent to njas
  Subject: NEW SEQUENCE FROM Jonathan Vos Post

 %I A000001
 %S A000001 2, 3, 23, 4, 235, 35, 26, 347, 237, 58,
 2359, 349, 2610, 311, 235711, 45712, 2313, 3813
 %N A000001 Replace n by the concatenation of its
 antidivisors.
 %C A000001 Number of antidivisors concatened to form
 a(n) is A066272(n). We may consider prime values of
 the concatenated antidivisor sequence, and we may
 iterate it, i.e. n, a(n), a(a(n)), a(a(a(n))) which
 leads to questions of trajectory, cycles, fixed
 points.
 %e A000001 Anti-divisors of 3 through 20:
 3: 2, so a(3) = 2.
 4: 3, so a(4) = 3.
 5: 2, 3, so a(5) = 23.
 6: 4, so a(6) = 4.
 7: 2, 3, 5, so a(7) = 235.
 17: 2, 3, 5, 7, 11, so a(17) = 235711
 %Y A000001 Cf. A037278, A066272, A120712, A106708,
 A130799.
 %O A000001 3
 %K A000001 ,base,easy,more,nonn,
 %A A000001 Jonathan Vos Post (jvospost2 at yahoo.com),
 Jul 20 2007
 RH
 RA 192.20.225.32




From maximilian.hasler at gmail.com  Sat Jul 21 02:38:38 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Fri, 20 Jul 2007 20:38:38 -0400
Subject: Concatenated anti-divisors
In-Reply-To: <5542af940707201535q1a3f22f5xfd0c2bdb892fab6d@mail.gmail.com>
References: <5542af940707201535q1a3f22f5xfd0c2bdb892fab6d@mail.gmail.com>
Message-ID: <3c3af2330707201738j2685797cr94e6ce49671e21c2@mail.gmail.com>

It appears to me that the definition

%C A066272 If an odd number d in the range 1 < d < n divides N where N
is any one of 2n-1, 2n or 2n+1 then N/d is called an anti-divisor of
n.

is not equivalent to your definition:

> Non-divisor: a number k which does not divide a given
> number x. Anti-divisor: a non-divisor k of x with the
> property that k is an odd divisor of 2x-1 or 2x+1, or
> an even divisor of 2x.

According to the first definition, n=3 cannot have an anti-divisor
since there is no odd d, 1<d<n.

M.H.




From maximilian.hasler at gmail.com  Sat Jul 21 03:35:14 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Fri, 20 Jul 2007 21:35:14 -0400
Subject: Concatenated anti-divisors
In-Reply-To: <3c3af2330707201738j2685797cr94e6ce49671e21c2@mail.gmail.com>
References: <5542af940707201535q1a3f22f5xfd0c2bdb892fab6d@mail.gmail.com>
	 <3c3af2330707201738j2685797cr94e6ce49671e21c2@mail.gmail.com>
Message-ID: <3c3af2330707201835n3b6b37c8r1d7594ea8514a132@mail.gmail.com>

It first appeared as if "your" definition (the second below) could be
found at least on some other web pages. Now I have big doubts. Is
there any *serious* reference for this notion ?
According to "your" definition,
antidiv( 3 ) = { 2, 5, 6, 7 }
since none of these numbers divide 3 and they are either even divisors
of 6 or odd divisors of 5 or 7.

Also, concerning the sequence :

A130799 Triangle read by rows in which row n (n>=3) list the anti-divisors of n.

I cannot see any triangular structure (cf.
http://www.research.att.com/~njas/sequences/table?a=130799
) and AFAICS the only way to find out where the anti-divisors of a
number n will start is to spot violations of strict increasing
monotony
(which is of course not failsafe,  e.g. if a number >2 would have no
antidivisors, or some number would have its largest antidiv. smaller
than the least antidiv of its successor....)

IMHO all these sequences need much clarification and cleanup of definitions.
Insofar more as the "authorative" definition seems to be on an
unavailable web page (and the version I can found via the "wayback"
machine does not contain a clear definition either - again several
"Or, to put it another way..." etc). All in all there is a heavy smell
of *very* original research in the air around this.

M.H.

On 7/20/07, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:
> It appears to me that the definition
>
> %C A066272 If an odd number d in the range 1 < d < n divides N where N
> is any one of 2n-1, 2n or 2n+1 then N/d is called an anti-divisor of
> n.
>
> is not equivalent to your definition:
>
> > Non-divisor: a number k which does not divide a given
> > number x. Anti-divisor: a non-divisor k of x with the
> > property that k is an odd divisor of 2x-1 or 2x+1, or
> > an even divisor of 2x.
>
> According to the first definition, n=3 cannot have an anti-divisor
> since there is no odd d, 1<d<n.
>
> M.H.
>




From maxale at gmail.com  Sat Jul 21 04:19:05 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Fri, 20 Jul 2007 19:19:05 -0700
Subject: Concatenated anti-divisors
In-Reply-To: <3c3af2330707201738j2685797cr94e6ce49671e21c2@mail.gmail.com>
References: <5542af940707201535q1a3f22f5xfd0c2bdb892fab6d@mail.gmail.com>
	 <3c3af2330707201738j2685797cr94e6ce49671e21c2@mail.gmail.com>
Message-ID: <d3dac270707201919v199594c7gab0cb2c18cee2340@mail.gmail.com>

There are glitches in both Jonathan's definition of anti-divisor and
the quoted comment to A066272.

Namely, Jonathan forgot to mention that k < x, and in the comment to
A066272 it should be "1 < d <= n" not "1 < d < n".

After such corrections these two definitions become equivalent and
consistent with the numerical values of A066272.

Max

On 7/20/07, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:
> It appears to me that the definition
>
> %C A066272 If an odd number d in the range 1 < d < n divides N where N
> is any one of 2n-1, 2n or 2n+1 then N/d is called an anti-divisor of
> n.
>
> is not equivalent to your definition:
>
> > Non-divisor: a number k which does not divide a given
> > number x. Anti-divisor: a non-divisor k of x with the
> > property that k is an odd divisor of 2x-1 or 2x+1, or
> > an even divisor of 2x.
>
> According to the first definition, n=3 cannot have an anti-divisor
> since there is no odd d, 1<d<n.




Dear seqfans, There are currently two versions
of the definition of anti-divisor in the OEIS:

%C A066272 If an odd number d in the range 1 < d < n divides N
where N is any one of 2n-1, 2n or 2n+1
then N/d is called an anti-divisor of n.

%e A066272 For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors > 1 {5,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 4.

But this definition fails for n = 3, as someone mentioned last night.
We know from the OEIS that 3 has a single antidivisor, 2. 
According to this definition 3 has no antidivisors.

There is also this program, which I have not checked:
%t A066272 antid[ n_ ] := Select[ Union[ Join[ Select[ Divisors[ 2n - 1 ], OddQ[ # ] && # != 1 & ], Select[ Divisors[ 2n + 1 ], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2*n ], OddQ[ # ] && # != 1 & ] ] ] }, # < n & ] ]; Table[ Length[ antid[ n ] ], {n, 1, 100} ]

The other definition is:

%I A130799
%S A130799 2,3,2,3,4,2,3,5,3,5,2,6,3,4,7,2,3,7,5,8,2,3,5,9,3,4,9,2,6,10,3,11,
%T A130799 2,3,5,7,11,4,5,7,12,2,3,13,3,8,13,2,6,14,3,4,5,9,15,2,3,5,9,15,7,
%U A130799 16,2,3,7,10,17,3,4,17,2,5,6,11,18,3,5,8,11,19,2,3,19,4,12,20,2,3,7
%N A130799 Triangle read by rows in which row n (n>=3) list the anti-divisors of n.
%C A130799 Non-divisor: a number k which does not divide a given number x. Anti-divisor: a non-divisor k of x with the property that k is an odd divisor of 2x-1 or 2x+1, or an even divisor of 2x.
%C A130799 There are no anti-divisors of 1 and 2.
%e A130799 Anti-divisors of 3 through 20:
%e A130799 3: 2
%e A130799 4: 3
%e A130799 5: 2, 3
%e A130799 6: 4
%e A130799 7: 2, 3, 5
%e A130799 8: 3, 5
%e A130799 9: 2, 6

This definition also fails for n = 3: it gives 5 antidivisors,
2,4,5,6,7.

The term anti-divisor seems to be due to Jon Perry.  
I wish I understood the motivation for the definition!

There are links to various webpages of his, but they are all broken
and he has not responded to my emails.

It seems to me that both of the above definitions are incorrect, and

where N is any one of 2n-1, 2n or 2n+1
then d = N/i is called an anti-divisor of n.

Equivalently, an anti-divisor of n is a number d in the range [1..n]
which does not divide n and is either an odd divisor of 2n-1 or 2n+1,
or an even divisor of 2n.

Now both definitions seem to work correctly for n=3, giving
a single anti-divisor, 2.

But I'm not too confident about all this - comments anyone?
As I said, I wish I understood the motivation for the definition!

Neil





From njas at research.att.com  Sat Jul 21 13:41:09 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Sat, 21 Jul 2007 07:41:09 -0400 (EDT)
Subject: definition of anti-divisor
Message-ID: <200707211141.l6LBf9dn795643@fry.research.att.com>

should be changed to:
Definition: If an odd number i in the range 1 < i <= n divides N
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Date: Sat, 21 Jul 2007 07:31:25 -0500
From: "Diana Mecum" <diana.mecum at gmail.com>
To: njas at research.att.com
Subject: Re: definition of anti-divisor
Cc: seqfan at ext.jussieu.fr
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Dr. Sloane,

As far as the second definition, and

> %C A130799 Non-divisor: a number k which does not divide a given number x.
Anti-divisor: a non-divisor k of x with the property that k is an odd
divisor of 2x-1 or 2x+1, or an even divisor of 2x.

> This definition also fails for n = 3: it gives 5 antidivisors,
2,4,5,6,7.

According to the second definition, above, an antidivisor must first be a
non-divisor of x. {4, 5, 6, 7} are not non-divisors of 3. Therefore, I
believe that the second definition is correct as it stands.

Diana

On 7/21/07, N. J. A. Sloane <njas at research.att.com> wrote:
>
>
> Dear seqfans, There are currently two versions
> of the definition of anti-divisor in the OEIS:
>
> %C A066272 If an odd number d in the range 1 < d < n divides N
> where N is any one of 2n-1, 2n or 2n+1
> then N/d is called an anti-divisor of n.
>
> %e A066272 For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd
> divisors > 1 {5,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the
> anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 4.
>
> But this definition fails for n = 3, as someone mentioned last night.
> We know from the OEIS that 3 has a single antidivisor, 2.
> According to this definition 3 has no antidivisors.
>
> There is also this program, which I have not checked:
> %t A066272 antid[ n_ ] := Select[ Union[ Join[ Select[ Divisors[ 2n - 1 ],
> OddQ[ # ] && # != 1 & ], Select[ Divisors[ 2n + 1 ], OddQ[ # ] && # != 1 &
> ], 2n/Select[ Divisors[ 2*n ], OddQ[ # ] && # != 1 & ] ] ] }, # < n & ] ];
> Table[ Length[ antid[ n ] ], {n, 1, 100} ]
>
> The other definition is:
>
> %I A130799
> %S A130799
> 2,3,2,3,4,2,3,5,3,5,2,6,3,4,7,2,3,7,5,8,2,3,5,9,3,4,9,2,6,10,3,11,
> %T A130799
> 2,3,5,7,11,4,5,7,12,2,3,13,3,8,13,2,6,14,3,4,5,9,15,2,3,5,9,15,7,
> %U A130799
> 16,2,3,7,10,17,3,4,17,2,5,6,11,18,3,5,8,11,19,2,3,19,4,12,20,2,3,7
> %N A130799 Triangle read by rows in which row n (n>=3) list the
> anti-divisors of n.
> %C A130799 Non-divisor: a number k which does not divide a given number x.
> Anti-divisor: a non-divisor k of x with the property that k is an odd
> divisor of 2x-1 or 2x+1, or an even divisor of 2x.
> %C A130799 There are no anti-divisors of 1 and 2.
> %e A130799 Anti-divisors of 3 through 20:
> %e A130799 3: 2
> %e A130799 4: 3
> %e A130799 5: 2, 3
> %e A130799 6: 4
> %e A130799 7: 2, 3, 5
> %e A130799 8: 3, 5
> %e A130799 9: 2, 6
>
> This definition also fails for n = 3: it gives 5 antidivisors,
> 2,4,5,6,7.
>
> The term anti-divisor seems to be due to Jon Perry.
> I wish I understood the motivation for the definition!
>
> There are links to various webpages of his, but they are all broken
> and he has not responded to my emails.
>
> It seems to me that both of the above definitions are incorrect, and
> should be changed to:
>
> Definition: If an odd number i in the range 1 < i <= n divides N
> where N is any one of 2n-1, 2n or 2n+1
> then d = N/i is called an anti-divisor of n.
>
> Equivalently, an anti-divisor of n is a number d in the range [1..n]
> which does not divide n and is either an odd divisor of 2n-1 or 2n+1,
> or an even divisor of 2n.
>
> Now both definitions seem to work correctly for n=3, giving
> a single anti-divisor, 2.
>
> But I'm not too confident about all this - comments anyone?
> As I said, I wish I understood the motivation for the definition!
>
> Neil
>
>


-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.

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Dr. Sloane,<br><br>As far as the second definition, and <br><br>&gt; %C A130799 Non-divisor: a number k which does not divide a given number
x. Anti-divisor: a non-divisor k of x with the property that k is an
odd divisor of 2x-1 or 2x+1, or an even divisor of 2x.<br><br>&gt; This definition also fails for n = 3: it gives 5 antidivisors,<br>2,4,5,6,7.<br><br>According to the second definition, above, an antidivisor must first be a non-divisor of x. {4, 5, 6, 7} are not non-divisors of 3. Therefore, I believe that the second definition is correct as it stands.
<br><br>Diana<br><br><div><span class="gmail_quote">On 7/21/07, <b class="gmail_sendername">N. J. A. Sloane</b> &lt;<a href="mailto:njas at research.att.com">njas at research.att.com</a>&gt; wrote:</span><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<br>Dear seqfans, There are currently two versions<br>of the definition of anti-divisor in the OEIS:<br><br>%C A066272 If an odd number d in the range 1 &lt; d &lt; n divides N<br>where N is any one of 2n-1, 2n or 2n+1<br>
then N/d is called an anti-divisor of n.<br><br>%e A066272 For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors &gt; 1 {5,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 4.
<br><br>But this definition fails for n = 3, as someone mentioned last night.<br>We know from the OEIS that 3 has a single antidivisor, 2.<br>According to this definition 3 has no antidivisors.<br><br>There is also this program, which I have not checked:
<br>%t A066272 antid[ n_ ] := Select[ Union[ Join[ Select[ Divisors[ 2n - 1 ], OddQ[ # ] &amp;&amp; # != 1 &amp; ], Select[ Divisors[ 2n + 1 ], OddQ[ # ] &amp;&amp; # != 1 &amp; ], 2n/Select[ Divisors[ 2*n ], OddQ[ # ] &amp;&amp; # != 1 &amp; ] ] ] }, # &lt; n &amp; ] ]; Table[ Length[ antid[ n ] ], {n, 1, 100} ]
<br><br>The other definition is:<br><br>%I A130799<br>%S A130799 2,3,2,3,4,2,3,5,3,5,2,6,3,4,7,2,3,7,5,8,2,3,5,9,3,4,9,2,6,10,3,11,<br>%T A130799 2,3,5,7,11,4,5,7,12,2,3,13,3,8,13,2,6,14,3,4,5,9,15,2,3,5,9,15,7,<br>%U A130799 16,2,3,7,10,17,3,4,17,2,5,6,11,18,3,5,8,11,19,2,3,19,4,12,20,2,3,7
<br>%N A130799 Triangle read by rows in which row n (n&gt;=3) list the anti-divisors of n.<br>%C A130799 Non-divisor: a number k which does not divide a given number x. Anti-divisor: a non-divisor k of x with the property that k is an odd divisor of 2x-1 or 2x+1, or an even divisor of 2x.
<br>%C A130799 There are no anti-divisors of 1 and 2.<br>%e A130799 Anti-divisors of 3 through 20:<br>%e A130799 3: 2<br>%e A130799 4: 3<br>%e A130799 5: 2, 3<br>%e A130799 6: 4<br>%e A130799 7: 2, 3, 5<br>%e A130799 8: 3, 5
<br>%e A130799 9: 2, 6<br><br>This definition also fails for n = 3: it gives 5 antidivisors,<br>2,4,5,6,7.<br><br>The term anti-divisor seems to be due to Jon Perry.<br>I wish I understood the motivation for the definition!
<br><br>There are links to various webpages of his, but they are all broken<br>and he has not responded to my emails.<br><br>It seems to me that both of the above definitions are incorrect, and<br>should be changed to:<br>
<br>Definition: If an odd number i in the range 1 &lt; i &lt;= n divides N<br>where N is any one of 2n-1, 2n or 2n+1<br>then d = N/i is called an anti-divisor of n.<br><br>Equivalently, an anti-divisor of n is a number d in the range [1..n]
<br>which does not divide n and is either an odd divisor of 2n-1 or 2n+1,<br>or an even divisor of 2n.<br><br>Now both definitions seem to work correctly for n=3, giving<br>a single anti-divisor, 2.<br><br>But I&#39;m not too confident about all this - comments anyone?
<br>As I said, I wish I understood the motivation for the definition!<br><br>Neil<br><br></blockquote></div><br><br clear="all"><br>-- <br>&quot;God made the integers, all else is the work of man.&quot; <br>L. Kronecker, Jahresber. DMV 2, S. 19.

------=_Part_148534_11559968.1185021085857--




From diana.mecum at gmail.com  Sat Jul 21 14:38:37 2007
From: diana.mecum at gmail.com (Diana Mecum)
Date: Sat, 21 Jul 2007 07:38:37 -0500
Subject: definition of anti-divisor
In-Reply-To: <a851af150707210531s2e01e2b2p3ed2e5c31e635d8d@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <a851af150707210531s2e01e2b2p3ed2e5c31e635d8d@mail.gmail.com>
Message-ID: <a851af150707210538y57a48ce0j32e9a952482038d2@mail.gmail.com>

Dr. Sloane,

As far as the second definition, and

> %C A130799 Non-divisor: a number k which does not divide a given number x.
Anti-divisor: a non-divisor k of x with the property that k is an odd
divisor of 2x-1 or 2x+1, or an even divisor of 2x.

> This definition also fails for n = 3: it gives 5 antidivisors,
2,4,5,6,7.

According to the second definition, above, an antidivisor must first be a
non-divisor of x. {4, 5, 6, 7} are not non-divisors of 3. Therefore, I
believe that the second definition is correct as it stands.

Diana

On 7/21/07, N. J. A. Sloane <njas at research.att.com> wrote:
>
>
> Dear seqfans, There are currently two versions
> of the definition of anti-divisor in the OEIS:
>
> %C A066272 If an odd number d in the range 1 < d < n divides N
> where N is any one of 2n-1, 2n or 2n+1
> then N/d is called an anti-divisor of n.
>
> %e A066272 For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd
> divisors > 1 {5,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the
> anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 4.
>
> But this definition fails for n = 3, as someone mentioned last night.
> We know from the OEIS that 3 has a single antidivisor, 2.
> According to this definition 3 has no antidivisors.
>
> There is also this program, which I have not checked:
> %t A066272 antid[ n_ ] := Select[ Union[ Join[ Select[ Divisors[ 2n - 1 ],
> OddQ[ # ] && # != 1 & ], Select[ Divisors[ 2n + 1 ], OddQ[ # ] && # != 1 &
> ], 2n/Select[ Divisors[ 2*n ], OddQ[ # ] && # != 1 & ] ] ] }, # < n & ] ];
> Table[ Length[ antid[ n ] ], {n, 1, 100} ]
>
> The other definition is:
>
> %I A130799
> %S A130799
> 2,3,2,3,4,2,3,5,3,5,2,6,3,4,7,2,3,7,5,8,2,3,5,9,3,4,9,2,6,10,3,11,
> %T A130799
> 2,3,5,7,11,4,5,7,12,2,3,13,3,8,13,2,6,14,3,4,5,9,15,2,3,5,9,15,7,
> %U A130799
> 16,2,3,7,10,17,3,4,17,2,5,6,11,18,3,5,8,11,19,2,3,19,4,12,20,2,3,7
> %N A130799 Triangle read by rows in which row n (n>=3) list the
> anti-divisors of n.
> %C A130799 Non-divisor: a number k which does not divide a given number x.
> Anti-divisor: a non-divisor k of x with the property that k is an odd
> divisor of 2x-1 or 2x+1, or an even divisor of 2x.
> %C A130799 There are no anti-divisors of 1 and 2.
> %e A130799 Anti-divisors of 3 through 20:
> %e A130799 3: 2
> %e A130799 4: 3
> %e A130799 5: 2, 3
> %e A130799 6: 4
> %e A130799 7: 2, 3, 5
> %e A130799 8: 3, 5
> %e A130799 9: 2, 6
>
> This definition also fails for n = 3: it gives 5 antidivisors,
> 2,4,5,6,7.
>
> The term anti-divisor seems to be due to Jon Perry.
> I wish I understood the motivation for the definition!
>
> There are links to various webpages of his, but they are all broken
> and he has not responded to my emails.
>
> It seems to me that both of the above definitions are incorrect, and
> should be changed to:
>
> Definition: If an odd number i in the range 1 < i <= n divides N
> where N is any one of 2n-1, 2n or 2n+1
> then d = N/i is called an anti-divisor of n.
>
> Equivalently, an anti-divisor of n is a number d in the range [1..n]
> which does not divide n and is either an odd divisor of 2n-1 or 2n+1,
> or an even divisor of 2n.
>
> Now both definitions seem to work correctly for n=3, giving
> a single anti-divisor, 2.
>
> But I'm not too confident about all this - comments anyone?
> As I said, I wish I understood the motivation for the definition!
>
> Neil
>
>


-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.

-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From maxale at gmail.com  Sat Jul 21 17:37:16 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Sat, 21 Jul 2007 08:37:16 -0700
Subject: definition of anti-divisor
In-Reply-To: <200707211141.l6LBf9dn795643@fry.research.att.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
Message-ID: <d3dac270707210837se098513xafd62ad62bf9360b@mail.gmail.com>

On 7/21/07, N. J. A. Sloane <njas at research.att.com> wrote:

> The term anti-divisor seems to be due to Jon Perry.
> I wish I understood the motivation for the definition!
>
> There are links to various webpages of his, but they are all broken
> and he has not responded to my emails.

Neil, I believe your corrections are correct ;)
Jon Perry's page on anti-divisors is available from archive.org:
http://web.archive.org/web/20070406031719/www.users.globalnet.co.uk/~perry/maths/antidivisor.htm

Max




When I posted my version of the correct definition
of anti-divisor earlier this morning, I should have
given credit to Maximilian Hasler and Max Alekseyev
for saying exactly the same thing in earlier
postings (which I had not read).
Maximilian, Max - my apologies!
Neil




From njas at research.att.com  Sat Jul 21 18:00:32 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Sat, 21 Jul 2007 12:00:32 -0400 (EDT)
Subject: credit for corrected definition of anti-divisor
In-Reply-To: <a851af150707210531s2e01e2b2p3ed2e5c31e635d8d@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <a851af150707210531s2e01e2b2p3ed2e5c31e635d8d@mail.gmail.com>
Message-ID: <200707211600.l6LG0WWW831020@fry.research.att.com>

> According to the second definition, above, an antidivisor must first be a
non-divisor of x. {4, 5, 6, 7} are not non-divisors of 3.

I was assuming that a non-divisor would be <= n by default.

Diana

On 7/21/07, Diana Mecum <diana.mecum at gmail.com> wrote:
>
> Dr. Sloane,
>
> As far as the second definition, and
>
> > %C A130799 Non-divisor: a number k which does not divide a given number
> x. Anti-divisor: a non-divisor k of x with the property that k is an odd
> divisor of 2x-1 or 2x+1, or an even divisor of 2x.
>
> > This definition also fails for n = 3: it gives 5 antidivisors,
> 2,4,5,6,7.
>
> According to the second definition, above, an antidivisor must first be a
> non-divisor of x. {4, 5, 6, 7} are not non-divisors of 3. Therefore, I
> believe that the second definition is correct as it stands.
>
> Diana
>
> On 7/21/07, N. J. A. Sloane <njas at research.att.com> wrote:
> >
> >
> > Dear seqfans, There are currently two versions
> > of the definition of anti-divisor in the OEIS:
> >
> > %C A066272 If an odd number d in the range 1 < d < n divides N
> > where N is any one of 2n-1, 2n or 2n+1
> > then N/d is called an anti-divisor of n.
> >
> > %e A066272 For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd
> > divisors > 1 {5,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the
> > anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 4.
> >
> > But this definition fails for n = 3, as someone mentioned last night.
> > We know from the OEIS that 3 has a single antidivisor, 2.
> > According to this definition 3 has no antidivisors.
> >
> > There is also this program, which I have not checked:
> > %t A066272 antid[ n_ ] := Select[ Union[ Join[ Select[ Divisors[ 2n - 1
> > ], OddQ[ # ] && # != 1 & ], Select[ Divisors[ 2n + 1 ], OddQ[ # ] && # != 1
> > & ], 2n/Select[ Divisors[ 2*n ], OddQ[ # ] && # != 1 & ] ] ] }, # < n & ] ];
> > Table[ Length[ antid[ n ] ], {n, 1, 100} ]
> >
> > The other definition is:
> >
> > %I A130799
> > %S A130799
> > 2,3,2,3,4,2,3,5,3,5,2,6,3,4,7,2,3,7,5,8,2,3,5,9,3,4,9,2,6,10,3,11,
> > %T A130799
> > 2,3,5,7,11,4,5,7,12,2,3,13,3,8,13,2,6,14,3,4,5,9,15,2,3,5,9,15,7,
> > %U A130799
> > 16,2,3,7,10,17,3,4,17,2,5,6,11,18,3,5,8,11,19,2,3,19,4,12,20,2,3,7
> > %N A130799 Triangle read by rows in which row n (n>=3) list the
> > anti-divisors of n.
> > %C A130799 Non-divisor: a number k which does not divide a given number
> > x. Anti-divisor: a non-divisor k of x with the property that k is an odd
> > divisor of 2x-1 or 2x+1, or an even divisor of 2x.
> > %C A130799 There are no anti-divisors of 1 and 2.
> > %e A130799 Anti-divisors of 3 through 20:
> > %e A130799 3: 2
> > %e A130799 4: 3
> > %e A130799 5: 2, 3
> > %e A130799 6: 4
> > %e A130799 7: 2, 3, 5
> > %e A130799 8: 3, 5
> > %e A130799 9: 2, 6
> >
> > This definition also fails for n = 3: it gives 5 antidivisors,
> > 2,4,5,6,7.
> >
> > The term anti-divisor seems to be due to Jon Perry.
> > I wish I understood the motivation for the definition!
> >
> > There are links to various webpages of his, but they are all broken
> > and he has not responded to my emails.
> >
> > It seems to me that both of the above definitions are incorrect, and
> > should be changed to:
> >
> > Definition: If an odd number i in the range 1 < i <= n divides N
> > where N is any one of 2n-1, 2n or 2n+1
> > then d = N/i is called an anti-divisor of n.
> >
> > Equivalently, an anti-divisor of n is a number d in the range [1..n]
> > which does not divide n and is either an odd divisor of 2n-1 or 2n+1,
> > or an even divisor of 2n.
> >
> > Now both definitions seem to work correctly for n=3, giving
> > a single anti-divisor, 2.
> >
> > But I'm not too confident about all this - comments anyone?
> > As I said, I wish I understood the motivation for the definition!
> >
> > Neil
> >
> >
>
>
> --
> "God made the integers, all else is the work of man."
> L. Kronecker, Jahresber. DMV 2, S. 19.




-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.
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From maxale at gmail.com  Sat Jul 21 22:17:53 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Sat, 21 Jul 2007 13:17:53 -0700
Subject: definition of anti-divisor
In-Reply-To: <200707211141.l6LBf9dn795643@fry.research.att.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
Message-ID: <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>

On 7/21/07, N. J. A. Sloane <njas at research.att.com> wrote:

> Definition: If an odd number i in the range 1 < i <= n divides N
> where N is any one of 2n-1, 2n or 2n+1
> then d = N/i is called an anti-divisor of n.
>
> Equivalently, an anti-divisor of n is a number d in the range [1..n]
> which does not divide n and is either an odd divisor of 2n-1 or 2n+1,
> or an even divisor of 2n.

Yet another definition:

k is a non-divisor of n iff:
1 < k < n
and
| (n mod k) - k/2 | <= 1.

Max




From maxale at gmail.com  Sat Jul 21 22:19:27 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Sat, 21 Jul 2007 13:19:27 -0700
Subject: definition of anti-divisor
In-Reply-To: <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
Message-ID: <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>

On 7/21/07, Max Alekseyev <maxale at gmail.com> wrote:

> Yet another definition:
>
> k is a non-divisor of n iff:
> 1 < k < n
> and
> | (n mod k) - k/2 | <= 1.

Oops. It should be

| (n mod k) - k/2 | < 1.

Max




From jvospost3 at gmail.com  Sun Jul 22 08:54:59 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Sat, 21 Jul 2007 23:54:59 -0700
Subject: definition of anti-divisor
In-Reply-To: <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
Message-ID: <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>

Thank you Neil, for putting the cached John Perry Page where we can
get it.  Thank you Max, Diana, et al for corrections and
clarifications.

To extend, then, eliminating the "more" (though anyone is welcome to
submit mathematica or equivalent, and a b-list):

COMMENT FROM Jonathan Vos Post RE A130846

%I A130846
%S A130846 2, 3, 23, 4, 235, 35, 26, 347, 237, 58, 2359, 349, 2610,
311, 235711, 45712, 2313, 3813, 2614, 345915, 235915, 716, 2371017,
3417, 2561118, 3581119, 2319, 41220, 237921, 35791321, 2561322, 3423,
23101423, 824, 2351525, 3457111525, 2671126, 391627, 23927, 45121728,
2351729, 3829, 26710131830, 3471331, 2351931, 51932,  239111433,
349112033
%N A130846 Replace n by the concatenation of its anti-divisors.
%H A130846 Jon Perry, <a
href="http://www.research.att.com/~njas/sequences/a066272a.html">The
Anti-Divisor</a>, cached copy.
%e A130846 a(21)-a(50) adapted from cached Perry page.
%Y A130846 Cf. A037278, A066272, A120712, A106708, A130799.
%O A130846 3
%K A130846 ,base,easy,nonn,
%A A130846 Jonathan Vos Post (jvospost2 at yahoo.com), Jul 22 2007

Next, the matters of iteration, fixed points, primes in A130846 can
commence, subject to Neil's taste as bounded by "less"itude.




From jvospost3 at gmail.com  Sun Jul 22 09:02:55 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Sun, 22 Jul 2007 00:02:55 -0700
Subject: definition of anti-divisor
In-Reply-To: <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
Message-ID: <5542af940707220002l4ca17d60i426f4a982d006d0b@mail.gmail.com>

Even the number of digits of, for n>2, A130846(n) = 1, 1, 2, 1, 3, 2,
2, 3, 3, 2, 4, 3, 4, 3, 6, 5, 4, 4, 4, 6, 6, 3... is nontrivial.

Raiding the spectre of indices n of record values of number of digits
of A130846(n).

I wonder if switching to base 2 or other bases is lessitudinal?  Or,
if tried, are there any interesting things to see?

What is the behavior of A130846(n)/n?

Since I don't really know what Jon Perry was getting at, I don't know
how dumb these questions and derived sequences are, hence I rely on
the judgment of seqfans.




From davidwwilson at comcast.net  Sun Jul 22 09:19:49 2007
From: davidwwilson at comcast.net (David Wilson)
Date: Sun, 22 Jul 2007 03:19:49 -0400
Subject: 919 conjecture
References: <200707171512.AA1076494578@TanyaKhovanova.com>
Message-ID: <004b01c7cc30$b15ae890$6501a8c0@yourxhtr8hvc4p>

The graph of A002375 strongly indicates that the number of Goldbach 
partitions of even n >= 4 remains positive. Similarly, the graph of A014085 
strongly indicates that the number of primes between adjacent positive 
squares remains positive. Though neither conjecture is proved, and a freak 
departure from the visible pattern is theoretically possible, the trends in 
the empirical evidence suggest that both conjectures are a very safe bet. 
Conway suggests calling such conjectures, with ample empirical evidence and 
little reason to doubt that evidence, "sureties". Thus Goldbach's and 
Legendre's conjectures are "sure" if not provably true.

I suggest that you count and graph the number of primes strictly between 
adjacent palindromes. I suspect that, except for anomalous adjacent 
palindrome pairs (10^k-1, 10^k+1) (which cannot be adjacent palindromic 
primes), the number of intervening primes will exhibit a visually convincing 
growth pattern that will convince you that (919, 929) is indeed the last 
pair of consecutive palindromes and consecutive primes.

----- Original Message ----- 
From: "Tanya Khovanova" <tanyakh at TanyaKhovanova.com>
To: <seqfan at ext.jussieu.fr>
Sent: Tuesday, July 17, 2007 6:12 PM
Subject: 919 conjecture


> Hello all,
>
> I was looking at A069803 - Smaller of two consecutive palindromic primes: 
> 2, 3, 5, 7, 181, 787, 919
> Conjectured to be complete.
>
> I am interested in seeing a proof that 919 is actually the largest 
> palindromic prime such that the next prime is palindromic.
> I checked up to 10^8 with Mathematica coding.
>
> Also, it is obvious that the distance from a palindrome n to the next one 
> is more than Sqrt(n/10). It is clear that prime gaps grow slower than 
> that. Looking at the prime gaps sequence A053303, it is easy to prove that 
> 919 is the last number like that up to 10^16.
>
> Is there a bound for prime gaps that proves that the gaps are less than 
> Sqrt(n/10) starting from some n?
>
> Tanya
>
>
> _________________________________________________________________
> Need personalized email and website? Look no further. It's easy
> with Doteasy $0 Web Hosting! Learn more at www.doteasy.com
>
>
> -- 
> No virus found in this incoming message.
> Checked by AVG Free Edition.
> Version: 7.5.476 / Virus Database: 269.10.8/906 - Release Date: 7/17/2007 
> 6:30 PM
> 





From pauldhanna at juno.com  Sun Jul 22 13:21:57 2007
From: pauldhanna at juno.com (Paul D. Hanna)
Date: Sun, 22 Jul 2007 07:21:57 -0400
Subject: A006336 - Unexpected Relation to Golden Ratio?
Message-ID: <20070722.072158.944.1.pauldhanna@juno.com>

Seqfans, 
     Consider the nice sequence A006336: 
a(n) = a(n-1) + a(n-1 - number of even terms so far). 
http://www.research.att.com/~njas/sequences/A006336
begins:
[1,2,3,5,8,11,16,21,29,40,51,67,88,109,138,167,207,258,309,376,...].
 
My COMMENT (NOT submitted to OEIS): 
-----------------------------------------------------------
It seems that A006336 can be generated by a rule using the golden ratio:
 
a(n) = a(n-1) + a([n/Phi]) for n>1 with a(1)=1  where Phi =
(sqrt(5)+1)/2, 
 
i.e., the number of even terms up to position n-1 equals: 
n-1 - [n/Phi] for n>1 where Phi = (sqrt(5)+1)/2. 
 
(PARI): 
a(n) = if(n==1,1, a(n-1) + a( floor(n/((sqrt(5)+1)/2)) )  )
-----------------------------------------------------------
 
Would someone verify if these are indeed equivalent definitions, at least
empirically?  
Or, what is the first position in which terms are NOT equal? 
 
If these are equivalent, then this is another unexpected appearance of
that ubiquitous constant. 
Thanks, 
     Paul 
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From tanyakh at TanyaKhovanova.com  Sun Jul 22 18:16:59 2007
From: tanyakh at TanyaKhovanova.com (Tanya Khovanova)
Date: Sun, 22 Jul 2007 09:16:59 -0700
Subject: sequence joke
In-Reply-To: <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
Message-ID: <200707220916.AA2126905490@TanyaKhovanova.com>

So, extending the table of my 20 July 2007 email:

===========
n  a(n)  factorization
3  2     prime
4  3     prime
5  23    prime
6  4     2^2
7  235   5 * 47
8  35    5 * 7
9  26    2 * 13
10 347   prime
11 237   3 * 79
12 58    2 * 29
13 2359  7 * 337
14 349   prime
15 2610  2 * 3^2 * 5 * 29
16 311   prime
17 235711  7 * 151 * 223
18 45712 2^4 * 2857
19 2313  3^2 * 257
20 3813  3 * 31 * 41
21 2614 2 * 1307
22 345915 3^2 * 5 * 7687
23 235915 5 * 29 * 1627
24 716   2^2 * 179
25 2371017 3 * 11 * 71849
26 3417 3 * 17 * 67
27 2561118 2 * 3 * 7 * 17^2 * 211
28 3581119 37 * 96787
29 2319  3 * 773
30 41220 2^2 * 3^2 * 5 * 229
31 237921 3 * 71 * 1117
32 35791321 37 * 967333
33 2561322 2 * 3 * 17 * 25111
34 3423  3 * 7 * 163
35 23101423 97 * 238159
36 824    2^3 * 103
37 2351525 5^2 * 11 * 17 * 503
38 3457111525 5^2 * 7 * 97 * 203659
39 2671126 2 * 1335563
40 391627 prime
41 23927   71 * 337
42 45121728 2^6 * 3 * 235009
43 2351729  17 * 138337
44 3829  7 * 547
45 26710131830 2 * 5 * 41^2 * 757 * 2099
46 3471331 prime
47 2351931 3 * 523 * 1499
48 51932 2^2 * 12983
49 239111433 3^4 * 11 * 43 * 79^2
50 349112033 12119 * 28807
51 2634  2 * 3 * 439
52 3578152135 5 * 79 * 9058613
53 2357152135 5 * 13 * 36263879
54 41236 2^2 * 13^2 * 61
55 23102237 73 * 316469
56 31637 17 * 1861
57 2562338 2 * 23 * 53 * 1051
58 3459132339 3 * 13 * 1229 * 72169
59 2379131739 3^2 * 877 * 301423
60 7811172440 2^3 * 5 * 195279311
61 231141 3 * 77047
62 3452541 3 * 1150847
...




Dear Seqfans,  My old friend Bernardo Recaman Santos just

%I A116700
%S A116700 12,21,23,31,32,34,41,42,43
%N A116700 "Early bird" numbers: write the natural numbers in a string 12345678910111213....
Sequence gives numbers which occur in the string ahead of their natural place.
%C A116700 Based on an idea by Argentinian puzzle creator Jaime Poniachik, these numbers were
MAA.
%e A116700 "12" appears at the start of the string, ahead of its position after "11", so is a
%K A116700 nonn,base,more,nice,new
%O A116700 1,1
%A A116700 Bernardo Recaman Santos (ignotus(AT)hotmail.com), Jul 22 2007

One needs to be a subscriber to have on-line access to
Math Horizons - can anyone get me a copy of the Martin Gardner
article?

Neil




From njas at research.att.com  Mon Jul 23 05:41:32 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Sun, 22 Jul 2007 23:41:32 -0400 (EDT)
Subject: Early Bird numbers
Message-ID: <200707230341.l6N3fWM9861587@fry.research.att.com>

sent a really nice sequence:
 introduced by Martin Gardner in the November 2005 issue of Math. Horizons, published by the
 member.
Return-Path: <warut822 at gmail.com>
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Message-ID: <482644420707230039r7e3eec65u2c9ec1956bf006e at mail.gmail.com>
Date: Mon, 23 Jul 2007 14:39:19 +0700
From: "Warut Roonguthai" <warut822 at gmail.com>
To: seqfan at ext.jussieu.fr
Subject: Re: Early Bird numbers
In-Reply-To: <200707230341.l6N3fWM9861587 at fry.research.att.com>
MIME-Version: 1.0
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X-Miltered: at shiva.jussieu.fr with ID 46A45B28.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr)!

Here are the first 157 terms of the early bird sequence:

12,21,23,31,32,34,41,42,43,45,51,52,53,54,56,61,62,63,64,65,
67,71,72,73,74,75,76,78,81,82,83,84,85,86,87,89,91,92,93,94,
95,96,97,98,99,101,110,111,112,121,122,123,131,132,141,142,
151,152,161,162,171,172,181,182,191,192,201,202,210,211,
212,213,214,215,216,217,218,219,220,221,222,223,231,232,
233,234,241,242,243,251,252,253,261,262,263,271,272,273,
281,282,283,291,292,293,301,302,303,310,311,312,313,314,
315,316,317,318,319,320,321,322,323,324,325,326,327,328,
329,330,331,332,333,334,341,342,343,344,345,351,352,353,
354,361,362,363,364,371,372,373,374,381,382,383,384,391,
392,393,394

Note that the natural place of n begins at

dn + 1 - (10^d - 1)/9

where d is the number of decimal digits of n, i.e.,

d = floor(log10(n)) + 1

Can this be a new sequence too?

BTW, I don't have access to Martin Gardner's article.

Warut

On 7/23/07, N. J. A. Sloane <njas at research.att.com> wrote:
>
> Dear Seqfans,  My old friend Bernardo Recaman Santos just
> sent a really nice sequence:
>
> %I A116700
> %S A116700 12,21,23,31,32,34,41,42,43
> %N A116700 "Early bird" numbers: write the natural numbers in a string 12345678910111213....
> Sequence gives numbers which occur in the string ahead of their natural place.
> %C A116700 Based on an idea by Argentinian puzzle creator Jaime Poniachik, these numbers were
>  introduced by Martin Gardner in the November 2005 issue of Math. Horizons, published by the
> MAA.
> %e A116700 "12" appears at the start of the string, ahead of its position after "11", so is a
>  member.
> %K A116700 nonn,base,more,nice,new
> %O A116700 1,1
> %A A116700 Bernardo Recaman Santos (ignotus(AT)hotmail.com), Jul 22 2007
>
> One needs to be a subscriber to have on-line access to
> Math Horizons - can anyone get me a copy of the Martin Gardner
> article?
>
> Neil
>




From jeremy.gardiner at btinternet.com  Mon Jul 23 09:45:37 2007
From: jeremy.gardiner at btinternet.com (JEREMY GARDINER)
Date: Mon, 23 Jul 2007 08:45:37 +0100 (BST)
Subject: Early Bird numbers
In-Reply-To: <200707230341.l6N3fWM9861587@fry.research.att.com>
Message-ID: <886534.93581.qm@web86607.mail.ukl.yahoo.com>

 
  I found some interesting references through Google:
   
  http://www.itsoc.org/publications/nltr/it1202.pdf
http://www.itsoc.org/publications/nltr/it0303web.pdf
http://membership.kcatm.org/pub/summ2-06.pdf
   
  Jeremy
   
  ----------------------------------------------------------------------------------------

"N. J. A. Sloane" <njas at research.att.com> wrote:
  
Dear Seqfans, My old friend Bernardo Recaman Santos just
sent a really nice sequence:

%I A116700
%S A116700 12,21,23,31,32,34,41,42,43
%N A116700 "Early bird" numbers: write the natural numbers in a string 12345678910111213....
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From warut822 at gmail.com  Mon Jul 23 14:35:22 2007
From: warut822 at gmail.com (Warut Roonguthai)
Date: Mon, 23 Jul 2007 19:35:22 +0700
Subject: Early Bird numbers
In-Reply-To: <450839.70489.qm@web86611.mail.ukl.yahoo.com>
References: <482644420707230039r7e3eec65u2c9ec1956bf006e@mail.gmail.com>
	 <450839.70489.qm@web86611.mail.ukl.yahoo.com>
Message-ID: <482644420707230535h41b06b78u11273f7c8804bf41@mail.gmail.com>

FYI, here's my Ubasic program for generating the early bird sequence:

10   X=""
20   for N=1 to 396
30   A=cutspc(str(N))
40   if instr(X,A)>0 then print N;
50   X+=A
60   next N

Warut




From jeremy.gardiner at btinternet.com  Mon Jul 23 13:53:35 2007
From: jeremy.gardiner at btinternet.com (JEREMY GARDINER)
Date: Mon, 23 Jul 2007 12:53:35 +0100 (BST)
Subject: Early Bird numbers
In-Reply-To: <482644420707230039r7e3eec65u2c9ec1956bf006e@mail.gmail.com>
Message-ID: <450839.70489.qm@web86611.mail.ukl.yahoo.com>

I checked the parity of Warut's early bird values and there *may* be a connection with the following sequences: I have no idea why this should be the case and my apology if this is a red herring !!!
   
  A091264  Matrix defined by a(n,k) = 2^n + (k-1), read by antidiagonals.
A128138  A000012 * A128132.
A128219  A000012 * A127701. a(1) = 1, a(2) = 2, a(3) = 2; by rows, n-1 terms of 2, 3, 4...followed by "n".
   
  Parity values below; note these fail to match in a couple of places, suggesting I may have made an editing error, or that perhaps Warut's values should be checked?
   
  Jeremy
           0,1,1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,
A091264 ,0,1,1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0, 
           1,0,0,0,1,1,0,1,0,0,0,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,
A128138 ,1,0,0,0,1,1,0,1,0,0,0,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0, 
A128219 ,1,0,0,0,1,1,0,1,0,0,0,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0, 
  A091264  Matrix defined by a(n,k) = 2^n + (k-1), read by antidiagonals.
A128138  A000012 * A128132.
A128219  A000012 * A127701. a(1) = 1, a(2) = 2, a(3) = 2; by rows, n-1 terms of 2, 3, 4...followed by "n".
   
  ------------------------------------------------------------------------------------------
Warut Roonguthai <warut822 at gmail.com> wrote:
  Here are the first 157 terms of the early bird sequence:

12,21,23,31,32,34,41,42,43,45,51,52,53,54,56,61,62,63,64,65,
67,71,72,73,74,75,76,78,81,82,83,84,85,86,87,89,91,92,93,94,
95,96,97,98,99,101,110,111,112,121,122,123,131,132,141,142,
151,152,161,162,171,172,181,182,191,192,201,202,210,211,
212,213,214,215,216,217,218,219,220,221,222,223,231,232,
233,234,241,242,243,251,252,253,261,262,263,271,272,273,
281,282,283,291,292,293,301,302,303,310,311,312,313,314,
315,316,317,318,319,320,321,322,323,324,325,326,327,328,
329,330,331,332,333,334,341,342,343,344,345,351,352,353,
354,361,362,363,364,371,372,373,374,381,382,383,384,391,
392,393,394

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From joshua.zucker at gmail.com  Mon Jul 23 15:58:50 2007
From: joshua.zucker at gmail.com (Joshua Zucker)
Date: Mon, 23 Jul 2007 06:58:50 -0700
Subject: Early Bird numbers
In-Reply-To: <482644420707230535h41b06b78u11273f7c8804bf41@mail.gmail.com>
References: <482644420707230039r7e3eec65u2c9ec1956bf006e@mail.gmail.com>
	 <450839.70489.qm@web86611.mail.ukl.yahoo.com>
	 <482644420707230535h41b06b78u11273f7c8804bf41@mail.gmail.com>
Message-ID: <721e81490707230658g33f6d925k8d0c7d0f7b80bf4e@mail.gmail.com>

I wrote my own program and let it run to make all the terms up to
1000.  Up to 394 they match the terms Warut's program produced.

--Joshua Zucker

12 21 23 31 32 34 41 42 43 45 51 52 53 54 56 61 62 63 64 65 67 71 72
73 74 75 76 78 81 82 83 84 85 86 87 89 91 92 93 94 95 96 97 98 99 101
110 111 112 121 122 123 131 132 141 142 151 152 161 162 171 172 181
182 191 192 201 202 210 211 212 213 214 215 216 217 218 219 220 221
222 223 231 232 233 234 241 242 243 251 252 253 261 262 263 271 272
273 281 282 283 291 292 293 301 302 303 310 311 312 313 314 315 316
317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333
334 341 342 343 344 345 351 352 353 354 361 362 363 364 371 372 373
374 381 382 383 384 391 392 393 394 401 402 403 404 410 411 412 413
414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430
431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 451 452
453 454 455 456 461 462 463 464 465 471 472 473 474 475 481 482 483
484 485 491 492 493 494 495 501 502 503 504 505 510 511 512 513 514
515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531
532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548
549 550 551 552 553 554 555 556 561 562 563 564 565 566 567 571 572
573 574 575 576 581 582 583 584 585 586 591 592 593 594 595 596 601
602 603 604 605 606 610 611 612 613 614 615 616 617 618 619 620 621
622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638
639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655
656 657 658 659 660 661 662 663 664 665 666 667 671 672 673 674 675
676 677 678 681 682 683 684 685 686 687 691 692 693 694 695 696 697
701 702 703 704 705 706 707 710 711 712 713 714 715 716 717 718 719
720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736
737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753
754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770
771 772 773 774 775 776 777 778 781 782 783 784 785 786 787 788 789
791 792 793 794 795 796 797 798 801 802 803 804 805 806 807 808 810
811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827
828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844
845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861
862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878
879 880 881 882 883 884 885 886 887 888 889 891 892 893 894 895 896
897 898 899 901 902 903 904 905 906 907 908 909 910 911 912 913 914
915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931
932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948
949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965
966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982
983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999

On 7/23/07, Warut Roonguthai <warut822 at gmail.com> wrote:
> FYI, here's my Ubasic program for generating the early bird sequence:
>
> 10   X=""
> 20   for N=1 to 396
> 30   A=cutspc(str(N))
> 40   if instr(X,A)>0 then print N;
> 50   X+=A
> 60   next N
>
> Warut
>




From maximilian.hasler at gmail.com  Mon Jul 23 18:00:24 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Mon, 23 Jul 2007 12:00:24 -0400
Subject: Early Bird numbers
In-Reply-To: <721e81490707230658g33f6d925k8d0c7d0f7b80bf4e@mail.gmail.com>
References: <482644420707230039r7e3eec65u2c9ec1956bf006e@mail.gmail.com>
	 <450839.70489.qm@web86611.mail.ukl.yahoo.com>
	 <482644420707230535h41b06b78u11273f7c8804bf41@mail.gmail.com>
	 <721e81490707230658g33f6d925k8d0c7d0f7b80bf4e@mail.gmail.com>
Message-ID: <3c3af2330707230900y56c82727tfae0564679ae8746@mail.gmail.com>

This is another example of how computers can denature things.
"Early birds" seem interesting when you do it by hand, for n<199, say
; but from then on it would be more interesting to study the
complementary sequence ("late birds" ? which appear not before their
"regular" place) - e.g. among 500..999 only about 50 terms are not
present.
At a first glance, I'd almost be tempted to conjecture that from a
certain rank on, the only late birds are of the form  d*10^k with d in
{1,...,9 }, and all other numbers are early birds...
M.H.

On 7/23/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
> I wrote my own program and let it run to make all the terms up to
> 1000.  Up to 394 they match the terms Warut's program produced.
>
> --Joshua Zucker
>
> 12 21 23 31 32 34 41 42 43 45 51 52 53 54 56 61 62 63 64 65 67 71 72
> 73 74 75 76 78 81 82 83 84 85 86 87 89 91 92 93 94 95 96 97 98 99 101
> 110 111 112 121 122 123 131 132 141 142 151 152 161 162 171 172 181
> 182 191 192 201 202 210 211 212 213 214 215 216 217 218 219 220 221
> 222 223 231 232 233 234 241 242 243 251 252 253 261 262 263 271 272
> 273 281 282 283 291 292 293 301 302 303 310 311 312 313 314 315 316
> 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333
> 334 341 342 343 344 345 351 352 353 354 361 362 363 364 371 372 373
> 374 381 382 383 384 391 392 393 394 401 402 403 404 410 411 412 413
> 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430
> 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 451 452
> 453 454 455 456 461 462 463 464 465 471 472 473 474 475 481 482 483
> 484 485 491 492 493 494 495 501 502 503 504 505 510 511 512 513 514
> 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531
> 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548
> 549 550 551 552 553 554 555 556 561 562 563 564 565 566 567 571 572
> 573 574 575 576 581 582 583 584 585 586 591 592 593 594 595 596 601
> 602 603 604 605 606 610 611 612 613 614 615 616 617 618 619 620 621
> 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638
> 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655
> 656 657 658 659 660 661 662 663 664 665 666 667 671 672 673 674 675
> 676 677 678 681 682 683 684 685 686 687 691 692 693 694 695 696 697
> 701 702 703 704 705 706 707 710 711 712 713 714 715 716 717 718 719
> 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736
> 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753
> 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770
> 771 772 773 774 775 776 777 778 781 782 783 784 785 786 787 788 789
> 791 792 793 794 795 796 797 798 801 802 803 804 805 806 807 808 810
> 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827
> 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844
> 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861
> 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878
> 879 880 881 882 883 884 885 886 887 888 889 891 892 893 894 895 896
> 897 898 899 901 902 903 904 905 906 907 908 909 910 911 912 913 914
> 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931
> 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948
> 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965
> 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982
> 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999
>
> On 7/23/07, Warut Roonguthai <warut822 at gmail.com> wrote:
> > FYI, here's my Ubasic program for generating the early bird sequence:
> >
> > 10   X=""
> > 20   for N=1 to 396
> > 30   A=cutspc(str(N))
> > 40   if instr(X,A)>0 then print N;
> > 50   X+=A
> > 60   next N
> >
> > Warut
> >
>




From warut822 at gmail.com  Mon Jul 23 18:10:00 2007
From: warut822 at gmail.com (Warut Roonguthai)
Date: Mon, 23 Jul 2007 23:10:00 +0700
Subject: A006336 - Unexpected Relation to Golden Ratio?
In-Reply-To: <20070722.072158.944.1.pauldhanna@juno.com>
References: <20070722.072158.944.1.pauldhanna@juno.com>
Message-ID: <482644420707230910s22b233aeg6b06cf2dcf0d2fe7@mail.gmail.com>

Very interesting observation, Paul. Your conjecture is too beautiful
to be wrong! However, it seems to be very difficult to prove. I've
checked the first 65,000 terms, but I think it would be easy for some
seqfan programmers to extend the calculation to 10,000,000 terms or so
with C or some other powerful tools. Note that we only have to keep
track of the parity of a(n), not its entire value.

Warut

On 7/22/07, Paul D. Hanna <pauldhanna at juno.com> wrote:
>
> Seqfans,
>      Consider the nice sequence A006336:
> a(n) = a(n-1) + a(n-1 - number of even terms so far).
> http://www.research.att.com/~njas/sequences/A006336
> begins:
> [1,2,3,5,8,11,16,21,29,40,51,67,88,109,138,167,207,258,309,376,...].
>
> My COMMENT (NOT submitted to OEIS):
> -----------------------------------------------------------
> It seems that A006336 can be generated by a rule using the golden ratio:
>
> a(n) = a(n-1) + a([n/Phi]) for n>1 with a(1)=1  where Phi = (sqrt(5)+1)/2,
>
> i.e., the number of even terms up to position n-1 equals:
> n-1 - [n/Phi] for n>1 where Phi = (sqrt(5)+1)/2.
>
> (PARI):
> a(n) = if(n==1,1, a(n-1) + a( floor(n/((sqrt(5)+1)/2)) )  )
> -----------------------------------------------------------
>
> Would someone verify if these are indeed equivalent definitions, at least
> empirically?
> Or, what is the first position in which terms are NOT equal?
>
> If these are equivalent, then this is another unexpected appearance of that
> ubiquitous constant.
> Thanks,
>      Paul




From Eric.Angelini at kntv.be  Mon Jul 23 18:25:48 2007
From: Eric.Angelini at kntv.be (Eric Angelini)
Date: Mon, 23 Jul 2007 18:25:48 +0200
Subject: Divisor d is the total number of divisors
Message-ID: <F05775148D000C4B9321A5113D53CB7D405F91@KNTV-SERVER01.kntv.local>



Hello SeqFans,

Is this seq of interest? 
If yes could someone check and compute a few more terms?

1,8,9,12,18,24,36,...

Integers I having one divisor which is also the total number of divisors of I.

 1 has 1 divisor which is 1
 8 has 4 divs and 4 is one of them
 9 has 3 divs and 3 is one of them
12 has 6 divs and 6 is one of them
18 has 6 divs and 6 is one of them
24 has 8 divs and 8 is one of them
36 has 9 divs and 9 is one of them
...

30 is not a member because 30 has 8 divs but not 8 itself : [1,2,3,5,6,10,15,30]

Best,
?.








From Eric.Angelini at kntv.be  Mon Jul 23 18:33:29 2007
From: Eric.Angelini at kntv.be (Eric Angelini)
Date: Mon, 23 Jul 2007 18:33:29 +0200
Subject: Divisor d is the total number of divisors
Message-ID: <F05775148D000C4B9321A5113D53CB7D405F92@KNTV-SERVER01.kntv.local>

 

Sorry, this is A033950
Best,
?.

(had forgotten "2" in my list -- no hit then in the OEIS :-(



-----Message d'origine-----
De : Eric Angelini 
Envoy? : lundi 23 juillet 2007 18:26
? : seqfan at ext.jussieu.fr
Objet : Divisor d is the total number of divisors



Hello SeqFans,

Is this seq of interest? 
If yes could someone check and compute a few more terms?

1,8,9,12,18,24,36,...

Integers I having one divisor which is also the total number of divisors of I.

 1 has 1 divisor which is 1
 8 has 4 divs and 4 is one of them
 9 has 3 divs and 3 is one of them
12 has 6 divs and 6 is one of them
18 has 6 divs and 6 is one of them
24 has 8 divs and 8 is one of them
36 has 9 divs and 9 is one of them
...

30 is not a member because 30 has 8 divs but not 8 itself : [1,2,3,5,6,10,15,30]

Best,
?.








From maximilian.hasler at gmail.com  Mon Jul 23 18:28:27 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Mon, 23 Jul 2007 12:28:27 -0400
Subject: Early Bird numbers
In-Reply-To: <3c3af2330707230900y56c82727tfae0564679ae8746@mail.gmail.com>
References: <482644420707230039r7e3eec65u2c9ec1956bf006e@mail.gmail.com>
	 <450839.70489.qm@web86611.mail.ukl.yahoo.com>
	 <482644420707230535h41b06b78u11273f7c8804bf41@mail.gmail.com>
	 <721e81490707230658g33f6d925k8d0c7d0f7b80bf4e@mail.gmail.com>
	 <3c3af2330707230900y56c82727tfae0564679ae8746@mail.gmail.com>
Message-ID: <3c3af2330707230928g2bbb5ce9u165e91d66b0e3eef@mail.gmail.com>

at the second glance, I take back my premature proposal concerning a
"late bird" conjecture - as shows the following PHP 1-liner,

Late birds:
<?php $s="."; for(; ++$i < 2000; $s .= $i) if( !strpos($s,"$i")) echo $i,", ";

there are many late birds in the "1xxxx" range for any number of digits.
Sorry...
M.H.

Late birds:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 22,
24, 25, 26, 27, 28, 29, 30, 33, 35, 36, 37, 38, 39, 40, 44, 46, 47,
48, 49, 50, 55, 57, 58, 59, 60, 66, 68, 69, 70, 77, 79, 80, 88, 90,
100, 102, 103, 104, 105, 106, 107, 108, 109, 113, 114, 115, 116, 117,
118, 119, 120, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136,
137, 138, 139, 140, 143, 144, 145, 146, 147, 148, 149, 150, 153, 154,
155, 156, 157, 158, 159, 160, 163, 164, 165, 166, 167, 168, 169, 170,
173, 174, 175, 176, 177, 178, 179, 180, 183, 184, 185, 186, 187, 188,
189, 190, 193, 194, 195, 196, 197, 198, 199, 200, 203, 204, 205, 206,
207, 208, 209, 224, 225, 226, 227, 228, 229, 230, 235, 236, 237, 238,
239, 240, 244, 245, 246, 247, 248, 249, 250, 254, 255, 256, 257, 258,
259, 260, 264, 265, 266, 267, 268, 269, 270, 274, 275, 276, 277, 278,
279, 280, 284, 285, 286, 287, 288, 289, 290, 294, 295, 296, 297, 298,
299, 300, 304, 305, 306, 307, 308, 309, 335, 336, 337, 338, 339, 340,
346, 347, 348, 349, 350, 355, 356, 357, 358, 359, 360, 365, 366, 367,
368, 369, 370, 375, 376, 377, 378, 379, 380, 385, 386, 387, 388, 389,
390, 395, 396, 397, 398, 399, 400, 405, 406, 407, 408, 409, 446, 447,
448, 449, 450, 457, 458, 459, 460, 466, 467, 468, 469, 470, 476, 477,
478, 479, 480, 486, 487, 488, 489, 490, 496, 497, 498, 499, 500, 506,
507, 508, 509, 557, 558, 559, 560, 568, 569, 570, 577, 578, 579, 580,
587, 588, 589, 590, 597, 598, 599, 600, 607, 608, 609, 668, 669, 670,
679, 680, 688, 689, 690, 698, 699, 700, 708, 709, 779, 780, 790, 799,
800, 809, 890, 900, 1000, 1002, 1003, 1004, 1005, 1006, 1007, 1008,
1009, 1010, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1019, 1020,
1022, 1023, ...

On 7/23/07, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:
> This is another example of how computers can denature things.
> "Early birds" seem interesting when you do it by hand, for n<199, say
> ; but from then on it would be more interesting to study the
> complementary sequence ("late birds" ? which appear not before their
> "regular" place) - e.g. among 500..999 only about 50 terms are not
> present.
> At a first glance, I'd almost be tempted to conjecture that from a
> certain rank on, the only late birds are of the form  d*10^k with d in
> {1,...,9 }, and all other numbers are early birds...
> M.H.
>
> On 7/23/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
> > I wrote my own program and let it run to make all the terms up to
> > 1000.  Up to 394 they match the terms Warut's program produced.
> >
> > --Joshua Zucker
> >
> > 12 21 23 31 32 34 41 42 43 45 51 52 53 54 56 61 62 63 64 65 67 71 72
> > 73 74 75 76 78 81 82 83 84 85 86 87 89 91 92 93 94 95 96 97 98 99 101
> > 110 111 112 121 122 123 131 132 141 142 151 152 161 162 171 172 181
> > 182 191 192 201 202 210 211 212 213 214 215 216 217 218 219 220 221
> > 222 223 231 232 233 234 241 242 243 251 252 253 261 262 263 271 272
> > 273 281 282 283 291 292 293 301 302 303 310 311 312 313 314 315 316
> > 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333
> > 334 341 342 343 344 345 351 352 353 354 361 362 363 364 371 372 373
> > 374 381 382 383 384 391 392 393 394 401 402 403 404 410 411 412 413
> > 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430
> > 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 451 452
> > 453 454 455 456 461 462 463 464 465 471 472 473 474 475 481 482 483
> > 484 485 491 492 493 494 495 501 502 503 504 505 510 511 512 513 514
> > 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531
> > 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548
> > 549 550 551 552 553 554 555 556 561 562 563 564 565 566 567 571 572
> > 573 574 575 576 581 582 583 584 585 586 591 592 593 594 595 596 601
> > 602 603 604 605 606 610 611 612 613 614 615 616 617 618 619 620 621
> > 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638
> > 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655
> > 656 657 658 659 660 661 662 663 664 665 666 667 671 672 673 674 675
> > 676 677 678 681 682 683 684 685 686 687 691 692 693 694 695 696 697
> > 701 702 703 704 705 706 707 710 711 712 713 714 715 716 717 718 719
> > 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736
> > 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753
> > 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770
> > 771 772 773 774 775 776 777 778 781 782 783 784 785 786 787 788 789
> > 791 792 793 794 795 796 797 798 801 802 803 804 805 806 807 808 810
> > 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827
> > 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844
> > 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861
> > 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878
> > 879 880 881 882 883 884 885 886 887 888 889 891 892 893 894 895 896
> > 897 898 899 901 902 903 904 905 906 907 908 909 910 911 912 913 914
> > 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931
> > 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948
> > 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965
> > 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982
> > 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999
> >
> > On 7/23/07, Warut Roonguthai <warut822 at gmail.com> wrote:
> > > FYI, here's my Ubasic program for generating the early bird sequence:
> > >
> > > 10   X=""
> > > 20   for N=1 to 396
> > > 30   A=cutspc(str(N))
> > > 40   if instr(X,A)>0 then print N;
> > > 50   X+=A
> > > 60   next N
> > >
> > > Warut
> > >
> >
>




From rayjchandler at sbcglobal.net  Mon Jul 23 18:31:23 2007
From: rayjchandler at sbcglobal.net (Ray Chandler)
Date: Mon, 23 Jul 2007 11:31:23 -0500
Subject: Divisor d is the total number of divisors
In-Reply-To: <F05775148D000C4B9321A5113D53CB7D405F91@KNTV-SERVER01.kntv.local>
References: <F05775148D000C4B9321A5113D53CB7D405F91@KNTV-SERVER01.kntv.local>
Message-ID: <021101c7cd46$e9b55b50$6600000a@HPm400y>

> 
> Is this seq of interest? 
> If yes could someone check and compute a few more terms?
> 
> 1,8,9,12,18,24,36,...
> > 
> Best,
> ?.
> 
Apparently a number of people thought so, see A033950.  (2 is also in the sequence).
Ray





>Is this seq of interest?
>If yes could someone check and compute a few more terms?
>
>1,8,9,12,18,24,36,...
>
>Integers I having one divisor which is also the total number of divisors of I.
>
> 1 has 1 divisor which is 1
> 8 has 4 divs and 4 is one of them
> 9 has 3 divs and 3 is one of them
>12 has 6 divs and 6 is one of them
>18 has 6 divs and 6 is one of them
>24 has 8 divs and 8 is one of them
>36 has 9 divs and 9 is one of them
>...
>
>30 is not a member because 30 has 8 divs but not 8 itself :
>[1,2,3,5,6,10,15,30]

See A033950.

Tony




From noe at sspectra.com  Mon Jul 23 18:30:49 2007
From: noe at sspectra.com (T. D. Noe)
Date: Mon, 23 Jul 2007 09:30:49 -0700
Subject: Divisor d is the total number of divisors
In-Reply-To:  <F05775148D000C4B9321A5113D53CB7D405F91@KNTV-SERVER01.kntv.local>
References:  <F05775148D000C4B9321A5113D53CB7D405F91@KNTV-SERVER01.kntv.local>
Message-ID: <p06240800c2ca880aa512@[192.168.1.43]>

------=_Part_151547_12232578.1185210638925
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Following Neil's suggestion to add the complement and his suggestion
for naming it "punctual birds" :

Subject: PRE-NUMBERED NEW SEQUENCE A131881 FROM Maximilian F. Hasler

%I A131881
%S A131881 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18,
19, 20, 22, 24, 25, 26, 27,
28, 29, 30, 33, 35, 36, 37, 38, 39, 40, 44, 46, 47, 48, 49, 50, 55,
57, 58, 59, 60,
66, 68, 69, 70, 77, 79, 80, 88, 90, 100, 102, 103, 104, 105, 106, 107,
108, 109, 113
%N A131881 "punctual birds" - numbers which are not in A116700
%C A131881 Numbers n that do not occur in the concatenation of 1,2,3...,n-1.
There is no punctual bird larger than 9*10^k and smaller than
10^(k+1), for any integer k.
%D A131881 Gardner, Martin. (November 2005). Transcendentals and early birds.
Math Horizons XIII(2), pp 5, 34.
%H A131881 <a href="http://www.itsoc.org/publications/nltr/it1202.pdf">Solomon
W. Golomb, "EARLY BIRD NUMBERS", in IEEE Information TheorySociety
Newsletter, Vol. 52, No. 4, December 2002</a>
%H A131881 <a href="http://membership.kcatm.org/pub/summ2-06.pdf">R.
Barger (editor), "Brain Teaser" in Volume 1, Issue 1 of KANSAS CITY
AREA TEACHERS OF MATHEMATICS"</a>
%e A131881 The first number not in this sequence is the early bird
"12" which occurs as concatenation of 1 and 2.
%o A131881 < ?php $s="."; for(; ++$i < 2000; $s .= $i )
if(!strpos($s,"$i")) echo $i,", ";
%Y A131881 Cf. A116700 (early birds)
%O A131881 1
%K A131881 ,base,easy,nonn,
%A A131881 Maximilian F. Hasler (Maximilian.Hasler at gmail.com), Jul 23 2007

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CjIyNjEgNTY5MAoyMjYyIDU2OTcKMjI2MyA1Njk4CjIyNjQgNTY5OQoyMjY1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CjIzOTYgNzA3OQoyMzk3IDcwODAKMjM5OCA3MDg4CjIzOTkgNzA4OQoyNDAw
IDcwOTAKMjQwMSA3MDk4CjI0MDIgNzA5OQoyNDAzIDc3NzkKMjQwNCA3Nzgw
CjI0MDUgNzc4OQoyNDA2IDc3OTAKMjQwNyA3Nzk5CjI0MDggNzgwMAoyNDA5
IDc4MDkKMjQxMCA3ODgwCjI0MTEgNzg4OQoyNDEyIDc4OTAKMjQxMyA3ODk5
CjI0MTQgNzkwMAoyNDE1IDc5MDkKMjQxNiA3OTc5CjI0MTcgNzk5MAoyNDE4
IDc5OTkKMjQxOSA4MDAwCjI0MjAgODAwOQoyNDIxIDgwODAKMjQyMiA4MDg5
CjI0MjMgODA5MAoyNDI0IDgwOTkKMjQyNSA4ODkwCjI0MjYgODkwMAoyNDI3
IDkwMDAKMjQyOCA5MDkwCg==

------=_Part_151547_12232578.1185210638925--




From keynews.tv at skynet.be  Mon Jul 23 19:17:28 2007
From: keynews.tv at skynet.be (Eric Angelini)
Date: Mon, 23 Jul 2007 19:17:28 +0200
Subject: Early Bird numbers
In-Reply-To: <721e81490707230658g33f6d925k8d0c7d0f7b80bf4e@mail.gmail.com>
Message-ID: <!&!AAAAAAAAAAAuAAAAAAAAACCO5uoaeU1MslQniZBHPHIBAPBXdRSNAAxLkyGlET1Ty30AAAADg9wAABAAAACvZHg3kvjSTaKgcanQ9RA1AQAAAAA=@skynet.be>


What about doing recursively the same with the "early bird" seq? 

We would then have to compare the concatenation of the "early bird" 
integers to 12345678910111213141516... and ask ourselves "what integers
come ahead now?"

By hand I get this (N = naturals, EB = Early Birds):

 
N=12345678910111213141516171819202122232425262728293031323334353637383940...
EB=122123313234414243455152535456616263646567717273747576788182838485868789.
..

EB(2)=12,13,14,21,22,23,24,25,26,31,32,33,34,35,36,37,38,41,42,43,44,45,
      46,47,48,51...

We could start again from there to compute EB(3), etc.

Could we bump into a "fixed point seq"? What would that mean?

Best,
?.



-----Message d'origine-----
De : Joshua Zucker [mailto:joshua.zucker at gmail.com] 
Envoy? : lundi 23 juillet 2007 15:59
? : Warut Roonguthai
Cc : seqfan at ext.jussieu.fr
Objet : Re: Early Bird numbers

I wrote my own program and let it run to make all the terms up to
1000.  Up to 394 they match the terms Warut's program produced.

--Joshua Zucker

12 21 23 31 32 34 41 42 43 45 51 52 53 54 56 61 62 63 64 65 67 71 72
73 74 75 76 78 81 82 83 84 85 86 87 89 91 92 93 94 95 96 97 98 99 101
110 111 112 121 122 123 131 132 141 142 151 152 161 162 171 172 181
182 191 192 201 202 210 211 212 213 214 215 216 217 218 219 220 221
222 223 231 232 233 234 241 242 243 251 252 253 261 262 263 271 272
273 281 282 283 291 292 293 301 302 303 310 311 312 313 314 315 316
317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333
334 341 342 343 344 345 351 352 353 354 361 362 363 364 371 372 373
374 381 382 383 384 391 392 393 394 401 402 403 404 410 411 412 413
414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430
431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 451 452
453 454 455 456 461 462 463 464 465 471 472 473 474 475 481 482 483
484 485 491 492 493 494 495 501 502 503 504 505 510 511 512 513 514
515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531
532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548
549 550 551 552 553 554 555 556 561 562 563 564 565 566 567 571 572
573 574 575 576 581 582 583 584 585 586 591 592 593 594 595 596 601
602 603 604 605 606 610 611 612 613 614 615 616 617 618 619 620 621
622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638
639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655
656 657 658 659 660 661 662 663 664 665 666 667 671 672 673 674 675
676 677 678 681 682 683 684 685 686 687 691 692 693 694 695 696 697
701 702 703 704 705 706 707 710 711 712 713 714 715 716 717 718 719
720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736
737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753
754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770
771 772 773 774 775 776 777 778 781 782 783 784 785 786 787 788 789
791 792 793 794 795 796 797 798 801 802 803 804 805 806 807 808 810
811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827
828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844
845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861
862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878
879 880 881 882 883 884 885 886 887 888 889 891 892 893 894 895 896
897 898 899 901 902 903 904 905 906 907 908 909 910 911 912 913 914
915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931
932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948
949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965
966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982
983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999

On 7/23/07, Warut Roonguthai <warut822 at gmail.com> wrote:
> FYI, here's my Ubasic program for generating the early bird sequence:
>
> 10   X=""
> 20   for N=1 to 396
> 30   A=cutspc(str(N))
> 40   if instr(X,A)>0 then print N;
> 50   X+=A
> 60   next N
>
> Warut
>






From maximilian.hasler at gmail.com  Mon Jul 23 19:17:23 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Mon, 23 Jul 2007 13:17:23 -0400
Subject: Early Bird numbers
In-Reply-To: <3c3af2330707231010g931f3cfl50118a3a07020f3b@mail.gmail.com>
References: <482644420707230039r7e3eec65u2c9ec1956bf006e@mail.gmail.com>
	 <450839.70489.qm@web86611.mail.ukl.yahoo.com>
	 <482644420707230535h41b06b78u11273f7c8804bf41@mail.gmail.com>
	 <721e81490707230658g33f6d925k8d0c7d0f7b80bf4e@mail.gmail.com>
	 <200707231623.l6NGNZMN1224353@fry.research.att.com>
	 <200707231659.l6NGxGe81231836@fry.research.att.com>
	 <3c3af2330707231010g931f3cfl50118a3a07020f3b@mail.gmail.com>
Message-ID: <3c3af2330707231017j1b99c155ga7514f3430c3b302@mail.gmail.com>

Definitely, I did too much administrative work this morning.
As the last line of my B-file shows, the comment is wrong.
Once again public apologies and Neil, please delete the 2nd line of the %C...
(9090 is a punctual bird.)
Maximilian

On 7/23/07, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:
> Following Neil's suggestion to add the complement and his suggestion
> for naming it "punctual birds" :
>
> Subject: PRE-NUMBERED NEW SEQUENCE A131881 FROM Maximilian F. Hasler
>
> %I A131881
> %S A131881 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18,
> 19, 20, 22, 24, 25, 26, 27,
> 28, 29, 30, 33, 35, 36, 37, 38, 39, 40, 44, 46, 47, 48, 49, 50, 55,
> 57, 58, 59, 60,
> 66, 68, 69, 70, 77, 79, 80, 88, 90, 100, 102, 103, 104, 105, 106, 107,
> 108, 109, 113
> %N A131881 "punctual birds" - numbers which are not in A116700
> %C A131881 Numbers n that do not occur in the concatenation of 1,2,3...,n-1.
> There is no punctual bird larger than 9*10^k and smaller than
> 10^(k+1), for any integer k.
> %D A131881 Gardner, Martin. (November 2005). Transcendentals and early birds.
> Math Horizons XIII(2), pp 5, 34.
> %H A131881 <a href="http://www.itsoc.org/publications/nltr/it1202.pdf">Solomon
> W. Golomb, "EARLY BIRD NUMBERS", in IEEE Information TheorySociety
> Newsletter, Vol. 52, No. 4, December 2002</a>
> %H A131881 <a href="http://membership.kcatm.org/pub/summ2-06.pdf">R.
> Barger (editor), "Brain Teaser" in Volume 1, Issue 1 of KANSAS CITY
> AREA TEACHERS OF MATHEMATICS"</a>
> %e A131881 The first number not in this sequence is the early bird
> "12" which occurs as concatenation of 1 and 2.
> %o A131881 < ?php $s="."; for(; ++$i < 2000; $s .= $i )
> if(!strpos($s,"$i")) echo $i,", ";
> %Y A131881 Cf. A116700 (early birds)
> %O A131881 1
> %K A131881 ,base,easy,nonn,
> %A A131881 Maximilian F. Hasler (Maximilian.Hasler at gmail.com), Jul 23 2007




From maxale at gmail.com  Mon Jul 23 19:29:13 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Mon, 23 Jul 2007 10:29:13 -0700
Subject: A006336 - Unexpected Relation to Golden Ratio?
In-Reply-To: <20070722.072158.944.1.pauldhanna@juno.com>
References: <20070722.072158.944.1.pauldhanna@juno.com>
Message-ID: <d3dac270707231029l69ce27b4kb5601a6491a064cb@mail.gmail.com>

On 7/22/07, Paul D. Hanna <pauldhanna at juno.com> wrote:
>
>
> Seqfans,
>      Consider the nice sequence A006336:
> a(n) = a(n-1) + a(n-1 - number of even terms so far).
> http://www.research.att.com/~njas/sequences/A006336
> begins:
> [1,2,3,5,8,11,16,21,29,40,51,67,88,109,138,167,207,258,309,376,...].
>
> My COMMENT (NOT submitted to OEIS):
> -----------------------------------------------------------
> It seems that A006336 can be generated by a rule using the golden ratio:
>
> a(n) = a(n-1) + a([n/Phi]) for n>1 with a(1)=1  where Phi = (sqrt(5)+1)/2,
>
>
> i.e., the number of even terms up to position n-1 equals:
> n-1 - [n/Phi] for n>1 where Phi = (sqrt(5)+1)/2.

To simplify notation, let p = Phi = (sqrt(5)+1)/2.

Lemma. The sets { [n*p] : n=1,2,3,... } and { [n*p^2] : n=1,2,3,... }
are disjoint, and every positive integer belongs to one (and only
one!) of these sets.
I leave the proof of this Lemma to the reader as a challenge.

Theorem. The number of even terms in A006336 up to position n-1 equals
n-1 - [n/p].

Proof by induction:

Suppose that the number of even terms in A006336(1..n) equal n -
[(n+1)/p] for every n=2..m. In other words, A006336(n) is even iff n -
[(n+1)/p] = (n-1 - [n/p])  + 1, that is equivalent to:
A006336(n) == [(n+1)/p] - [n/p]  (mod 2).

We will prove that the same statement is true for n=m+1.

By the definition of A006336 and our induction hypothesis, we have
a(m+1) = a(m) + a([(m+1)/p]) == [(m+1)/p] - [m/p] + [([(m+1)/p]+1)/p]
- [[(m+1)/p]/p] (mod 2).
Therefore, we need to prove that
[(m+2)/p] - [(m+1)/p] == [(m+1)/p] - [m/p] + [([(m+1)/p]+1)/p] -
[[(m+1)/p]/p] (mod 2)
or
[(m+2)/p] + [m/p] + [([(m+1)/p]+1)/p] + [[(m+1)/p]/p] == 0 (mod 2).

Let m+1 = q*p + r, where q is integer and 0 < r < p, and q = s*p + t,
where s is integer and 0 < t < p. Then m+1 = (s*p + t)*p + r = s*p^2 +
t*p + r.

It is easy to see that [(m+2)/p] + [m/p] = 0 (mod 2) if and only if
p-1 < r < 1.
Similarly, [([(m+1)/p]+1)/p] + [[(m+1)/p]/p] == 0 (mod 2) if and only
if t < p-1.

There are three cases when [(m+2)/p] + [m/p] and [([(m+1)/p]+1)/p] +
[[(m+1)/p]/p] may have different oddness:

1) If p-1 < r < 1 and t > p-1 then m = [q*p] and m+1 = [(q+1)*p].
We also have
m+1 = s*p^2 + t*p + r > s*p^2 + (p-1)*p + p - 1 = (s+1)*p^2 - 1
and
m+1 = s*p^2 + t*p + r < s*p^2 + p*p + 1 = (s+1)*p^2 + 1,
implying that [(s+1)*p^2] = m+1 or [(s+1)*p^2] = m, a contradiction to Lemma.

2) If t < p-1 and r < p-1 then m = [q*p] and m+2 = [(q+1)*p].
We also have
m+1 = s*p^2 + t*p + r > s*p^2
and
m+1 = s*p^2 + t*p + r < s*p^2 + (p-1)*p + p-1 = (s+1)*p^2 - 1,
implying that either [s*p^2] = m or [(s+1)*p^2] = m+2, a contradiction to Lemma.

3) If t < p-1 and r > 1 then m-1 = [q*p], m+1 = [(q+1)*p].
We also have
m+1 = s*p^2 + t*p + r > s*p^2 + 1
and
m+1 = s*p^2 + t*p + r < s*p^2 + (p-1)*p + p = (s+1)*p^2
implying that either [s*p^2] = m-1 or [(s+1)*p^2] = m+1, a
contradiction to Lemma.

Therefore, [(m+2)/p] + [m/p] = 0 (mod 2) if and only if [(m+2)/p] +
[m/p] = 0 (mod 2), implying that [(m+2)/p] + [m/p] + [([(m+1)/p]+1)/p]
+ [[(m+1)/p]/p] == 0 (mod 2).

Q.E.D.

Max




From davidwwilson at comcast.net  Mon Jul 23 19:56:08 2007
From: davidwwilson at comcast.net (David Wilson)
Date: Mon, 23 Jul 2007 13:56:08 -0400
Subject: Divisor d is the total number of divisors
References: <F05775148D000C4B9321A5113D53CB7D405F91@KNTV-SERVER01.kntv.local>
Message-ID: <002c01c7cd52$bf849ce0$6501a8c0@yourxhtr8hvc4p>

These are the refactorable numbers, A033950.

----- Original Message ----- 
From: "Eric Angelini" <Eric.Angelini at kntv.be>
To: <seqfan at ext.jussieu.fr>
Sent: Monday, July 23, 2007 12:25 PM
Subject: Divisor d is the total number of divisors


>
>
> Hello SeqFans,
>
> Is this seq of interest?
> If yes could someone check and compute a few more terms?
>
> 1,8,9,12,18,24,36,...
>
> Integers I having one divisor which is also the total number of divisors 
> of I.
>
> 1 has 1 divisor which is 1
> 8 has 4 divs and 4 is one of them
> 9 has 3 divs and 3 is one of them
> 12 has 6 divs and 6 is one of them
> 18 has 6 divs and 6 is one of them
> 24 has 8 divs and 8 is one of them
> 36 has 9 divs and 9 is one of them
> ...
>
> 30 is not a member because 30 has 8 divs but not 8 itself : 
> [1,2,3,5,6,10,15,30]
>
> Best,
> ?.
>
>
>
>
>
>
> -- 
> No virus found in this incoming message.
> Checked by AVG Free Edition.
> Version: 7.5.476 / Virus Database: 269.10.14/912 - Release Date: 7/22/2007 
> 7:02 PM
> 




At 10:40 AM -0700 7/23/07, Max Alekseyev wrote:
>On 7/22/07, Paul D. Hanna <pauldhanna at juno.com> wrote:
>>
>>
>> Seqfans,
>>      Consider the nice sequence A006336:
>> a(n) = a(n-1) + a(n-1 - number of even terms so far).
>> http://www.research.att.com/~njas/sequences/A006336
>> begins:
>> [1,2,3,5,8,11,16,21,29,40,51,67,88,109,138,167,207,258,309,376,...].
>>
>> My COMMENT (NOT submitted to OEIS):
>> -----------------------------------------------------------
>> It seems that A006336 can be generated by a rule using the golden ratio:
>>
>> a(n) = a(n-1) + a([n/Phi]) for n>1 with a(1)=1  where Phi = (sqrt(5)+1)/2,
>>
>>
>> i.e., the number of even terms up to position n-1 equals:
>> n-1 - [n/Phi] for n>1 where Phi = (sqrt(5)+1)/2.
>
>To simplify notation, let p = Phi = (sqrt(5)+1)/2.

Nice, Max.

What does this sequence count? It is similar to A00123 and A005704, which
both have a recursion a(n)=a(n-1)+a([n/k]), where k is 2 and 3,
respectively.  Those sequences count "number of partitions of k*n into
powers of k". For sequence A006336, k=Phi.  Does A006336(n) count the
number of partitions of n*Phi into powers of Phi?

Tony





From noe at sspectra.com  Mon Jul 23 20:39:32 2007
From: noe at sspectra.com (T. D. Noe)
Date: Mon, 23 Jul 2007 11:39:32 -0700
Subject: A006336 - Unexpected Relation to Golden Ratio?
In-Reply-To: <d3dac270707231029l69ce27b4kb5601a6491a064cb@mail.gmail.com>
References: <20070722.072158.944.1.pauldhanna@juno.com>
 <d3dac270707231029l69ce27b4kb5601a6491a064cb@mail.gmail.com>
Message-ID: <p06240804c2caa4c86185@[192.168.1.43]>

Dear seqfans,

I#ve been surprised not to find sequences of the following form in the OEIS:

a(n)=min(k in N: sigma(r,n)=sigma(r,k)) with sigma(r,n)=sum of the r-th
power of the divisors of n:

new[r_, n_] :=
  (If[Head[#1] === rep, #1 = n, #1] & )[rep[DivisorSigma[r, n]]]

for r=0 (number of divisors)

Clear[rep];
(new[0, #1] & ) /@ Range[0, 100]

{0, 1, 2, 2, 4, 2, 6, 2, 6, 4, 6, 2, 12, 2, 6, 6, 16, 2, 12, 2, 12,
  6, 6, 2, 24, 4, 6, 6, 12, 2, 24, 2, 12, 6, 6, 6, 36, 2, 6, 6, 24,
  2, 24, 2, 12, 12, 6, 2, 48, 4, 12, 6, 12, 2, 24, 6, 24, 6, 6, 2,
  60, 2, 6, 12, 64, 6, 24, 2, 12, 6, 24, 2, 60, 2, 6, 12, 12, 6,
  24, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24, 2, 60, 6, 12, 6, 6, 6, 60,
  2, 12, 12, 36}

for r=1 (sum of divisors)

Clear[rep];
(new[1, #1] & ) /@ Range[0, 100]

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 6, 12, 13, 14, 14, 16, 10, 18,
  19, 20, 21, 22, 14, 24, 16, 20, 27, 28, 29, 30, 21, 32, 33, 34,
  33, 36, 37, 24, 28, 40, 20, 42, 43, 44, 45, 30, 33, 48, 49, 50,
  30, 52, 34, 54, 30, 54, 57, 40, 24, 60, 61, 42, 63, 64, 44, 66,
  67, 68, 42, 66, 30, 72, 73, 74, 48, 76, 42, 60, 57, 80, 81, 68,
  44, 84, 85, 86, 54, 88, 40, 90, 91, 60, 93, 66, 54, 96, 52, 98,
  99, 100}

r=2 (sum of squares of divisors)

Clear[rep];
(new[2, #1] & ) /@ Range[0, 100]

{0, 1, 2, 3, 4, 5, 6, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
  19, 20, 21, 22, 23, 24, 25, 24, 27, 28, 29, 30, 31, 32, 33, 34,
  30, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 40, 48, 49, 50,
  51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,
  67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 66, 78, 79, 80, 81, 82,
  83, 84, 85, 86, 87, 88, 89, 90, 78, 92, 93, 94, 95, 96, 97, 98,
  99, 100}

and so on.

Are these of interest? And if so, up to which exponent r?

Peter



>What does this sequence count? It is similar to A00123 and A005704, which
>both have a recursion a(n)=a(n-1)+a([n/k]), where k is 2 and 3,
>respectively.  Those sequences count "number of partitions of k*n into
>powers of k". For sequence A006336, k=Phi.  Does A006336(n) count the
>number of partitions of n*Phi into powers of Phi?

Answering my own question.

If the recursion starts with a(0)=1, then I think we obtain "number of
partitions of n*Phi into powers of Phi".  That sequence is 1, 2, 4, 6, 10,
16, 22,..., which I just submitted as A131882.

We need negative powers of Phi also, letting p=Phi and q=1/Phi

n=0:	0*p = {}				for 1 partition

n=1:	1*p = p = 1+q				for 2 partitions

n=2:	2*p = p+p = 1+p+q = 1+1+q+q = p^2+q	for 4 partitions

etc. tedious!

So A006336(n), which starts with a(1)=1, counts 1/2 of the "number of
partitions of n*Phi into powers of Phi"

Tony




From noe at sspectra.com  Mon Jul 23 21:39:53 2007
From: noe at sspectra.com (T. D. Noe)
Date: Mon, 23 Jul 2007 12:39:53 -0700
Subject: A006336 - Unexpected Relation to Golden Ratio?
In-Reply-To: <p06240804c2caa4c86185@[192.168.1.43]>
References: <20070722.072158.944.1.pauldhanna@juno.com>
 <d3dac270707231029l69ce27b4kb5601a6491a064cb@mail.gmail.com>
 <p06240804c2caa4c86185@[192.168.1.43]>
Message-ID: <p06240806c2cab0c23025@[192.168.1.43]>

------=_Part_156860_32560920.1185232257325
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Content-Transfer-Encoding: 7bit
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concerning the "duplicates"

http://www.research.att.com/~njas/sequences/?q=id:A087186|id:A087189

I think all terms of the second one are wrong.

A087189(1) should be = (A005191(3)-1)/2/3^2 = 1
A087189(2) should be = (A005191(7)-1)/2/7^2 = 83
etc, instead of what is listed :
A087189 2,4,31,304,40044,500522,86094668,1167752848,225001039696,

Thus these are no duplicates at all. PARI code :

A087189(n) = local(p=prime(n+2-(n==1))); (A005191(p)-1)/2/p^2
A005191(n) = sum(k=0,(2*n)\5,binomial(n,k)*binomial(-n,2*n-5*k))

attached both sequences up to n=100 ; below the corrected entry for
the wrong one.

M.H.

%I A087189
%S A087189 1, 83, 16907, 279021, 89444018, 1695011087, 658067703933,
5768410509553937, 122108313460051500, 1226978854222034501448,
593538703869555995238872, 13175226571428140572866093,
6594852118968926152838341468, 76339779380132942579800334346403
%N A087189 (P(p)-1)/2/p^2 where p runs through the odd primes
different from 5 and P(k) is
               the k-th central pentanomial coefficient (A005191).
%o A087189 (PARI) A087189(n) = local(p=prime(n+2-(n==1)));
(A005191(p)-1)/2/p^2 \\ - M.F. Hasler
%Y A087189 Adjacent sequences: A087186 A087187 A087188 this_sequence
A087190 A087191 A087192
%Y A087189 Sequence in context: A018294 A051569 A087186 this_sequence
A051759 A051570 A082913
%K A087189 nonn
%O A087189 1,1
%A A087189 Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 19 2003
%E A087189 Corrected and extended by Maximilian F. Hasler
(Maximilian.Hasler at gmail.com), Jul 23 2007

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NDg1ODA0NjU0NTgxMwo=

------=_Part_156860_32560920.1185232257325--




From maximilian.hasler at gmail.com  Tue Jul 24 01:37:12 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Mon, 23 Jul 2007 19:37:12 -0400
Subject: Outstanding duplicates
In-Reply-To: <46a5200f.17bb720a.66c6.ffffffabSMTPIN_ADDED@mx.google.com>
References: <46a5200f.17bb720a.66c6.ffffffabSMTPIN_ADDED@mx.google.com>
Message-ID: <3c3af2330707231637i456739e7u74d5b064e8c5135@mail.gmail.com>

These :

http://www.research.att.com/~njas/sequences/?q=id:A023237|id:A105434

can be moved to the section "STRAIGHTFORWARD DUPES":

A023237  Numbers n such that n and 10n + 1 both prime.  	
A105434  Primes which with a 1 appended stay prime.

obviously the same.

M.H.




From alec at mihailovs.com  Tue Jul 24 02:52:14 2007
From: alec at mihailovs.com (Alec Mihailovs)
Date: Mon, 23 Jul 2007 19:52:14 -0500
Subject: A006336 - Unexpected Relation to Golden Ratio?
In-Reply-To: <d3dac270707231029l69ce27b4kb5601a6491a064cb@mail.gmail.com>
References: <20070722.072158.944.1.pauldhanna@juno.com> <d3dac270707231029l69ce27b4kb5601a6491a064cb@mail.gmail.com>
Message-ID: <1846E28BE6ED4805A69C8A9D6954FEA6@AlecPC>

From: "Max Alekseyev" <maxale at gmail.com>
Sent: Monday, July 23, 2007 12:29 PM

> On 7/22/07, Paul D. Hanna <pauldhanna at juno.com> wrote:
>>      Consider the nice sequence A006336:
>> a(n) = a(n-1) + a(n-1 - number of even terms so far).
>> http://www.research.att.com/~njas/sequences/A006336
>> -----------------------------------------------------------
>> It seems that A006336 can be generated by a rule using the golden ratio:
>> a(n) = a(n-1) + a([n/Phi]) for n>1 with a(1)=1  where Phi = 
>> (sqrt(5)+1)/2,
>
> Lemma. The sets { [n*p] : n=1,2,3,... } and { [n*p^2] : n=1,2,3,... }
> are disjoint, and every positive integer belongs to one (and only
> one!) of these sets.
> I leave the proof of this Lemma to the reader as a challenge.
>
> Theorem. The number of even terms in A006336 up to position n-1 equals
> n-1 - [n/p].

Another proof is based on the fact that A006336 mod 2 is A005614.

The proof is rather simple, but I don't have time to write it in a simple 
way at the moment.

I hope that Max can write a simple version of it based on his proof of the 
Theorem above.

By the way, looking at that, and searching for the initial terms of A001950, 
I found that A090909 seems to be a duplicate of A001950. Again, I don't have 
much of free time at the moment and I live it to Max to prove that.

Alec 





From jvospost3 at gmail.com  Tue Jul 24 03:11:31 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Mon, 23 Jul 2007 18:11:31 -0700
Subject: definition of anti-divisor
In-Reply-To: <5542af940707231709n330ca0ffkc581bb34984b1154@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
	 <5542af940707221117t4e411e40o840c9228f2e9b7f8@mail.gmail.com>
	 <3c3af2330707231221y5690ec3cm473f38b5f2bc6adc@mail.gmail.com>
	 <5542af940707231709n330ca0ffkc581bb34984b1154@mail.gmail.com>
Message-ID: <5542af940707231811t4044fa14rcdc33c9a227a9562@mail.gmail.com>

Any work since 2001 on whether or not there is a 6th anti-prime?  How
far has this been searched?  Any proofs or disproofs as to finiteness
of A066466?

COMMENT FROM Jonathan Vos Post RE A066466

%I A066466
%S A066466 3, 4, 6, 96, 393216
%N A066466 Numbers having just one anti-divisor.
%C A066466 Jon Perry calls these anti-prime numbers, saying that these
are the only 5 known. This sequence is worth extending, if possible,
or proving finite.
%F A066466 A066272(a(n)) = 1.
%Y A066466 Cf. A066272.
%O A066466 1
%K A066466 ,more,nonn,
%A A066466 Jonathan Vos Post (jvospost2 at yahoo.com), Jul 23 2007




From alec at mihailovs.com  Tue Jul 24 03:16:57 2007
From: alec at mihailovs.com (Alec Mihailovs)
Date: Mon, 23 Jul 2007 20:16:57 -0500
Subject: A006336 - Unexpected Relation to Golden Ratio?
In-Reply-To: <1846E28BE6ED4805A69C8A9D6954FEA6@AlecPC>
References: <20070722.072158.944.1.pauldhanna@juno.com> <d3dac270707231029l69ce27b4kb5601a6491a064cb@mail.gmail.com> <1846E28BE6ED4805A69C8A9D6954FEA6@AlecPC>
Message-ID: <707C3179321F466FAC06D9EDA30EF672@AlecPC>

I forgot to mention that 

n-1- (number of even terms thus far) = number of odd terms thus far 

and 

number of odd terms = the sum of the terms 

in a 0-1 sequence. 

Alec




From maxale at gmail.com  Tue Jul 24 03:57:27 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Mon, 23 Jul 2007 18:57:27 -0700
Subject: definition of anti-divisor
In-Reply-To: <5542af940707231811t4044fa14rcdc33c9a227a9562@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
	 <5542af940707221117t4e411e40o840c9228f2e9b7f8@mail.gmail.com>
	 <3c3af2330707231221y5690ec3cm473f38b5f2bc6adc@mail.gmail.com>
	 <5542af940707231709n330ca0ffkc581bb34984b1154@mail.gmail.com>
	 <5542af940707231811t4044fa14rcdc33c9a227a9562@mail.gmail.com>
Message-ID: <d3dac270707231857y6dae1f1cq27367b56568301cb@mail.gmail.com>

Except element 4, the elements of A066466 have form 2^k*p where p is
odd prime and both 2^(k+1)*p-1, 2^(k+1)*p+1 are prime (i.e., twin
primes).
In other words, A066466 without element 4 is a subsequence of A040040,
containing elements of the form 2^k*p with prime p.

Max

On 7/23/07, Jonathan Post <jvospost3 at gmail.com> wrote:
> Any work since 2001 on whether or not there is a 6th anti-prime?  How
> far has this been searched?  Any proofs or disproofs as to finiteness
> of A066466?
>
> COMMENT FROM Jonathan Vos Post RE A066466
>
> %I A066466
> %S A066466 3, 4, 6, 96, 393216
> %N A066466 Numbers having just one anti-divisor.
> %C A066466 Jon Perry calls these anti-prime numbers, saying that these
> are the only 5 known. This sequence is worth extending, if possible,
> or proving finite.
> %F A066466 A066272(a(n)) = 1.
> %Y A066466 Cf. A066272.
> %O A066466 1
> %K A066466 ,more,nonn,
> %A A066466 Jonathan Vos Post (jvospost2 at yahoo.com), Jul 23 2007
>




From maxale at gmail.com  Tue Jul 24 04:29:49 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Mon, 23 Jul 2007 19:29:49 -0700
Subject: definition of anti-divisor
In-Reply-To: <d3dac270707231857y6dae1f1cq27367b56568301cb@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
	 <5542af940707221117t4e411e40o840c9228f2e9b7f8@mail.gmail.com>
	 <3c3af2330707231221y5690ec3cm473f38b5f2bc6adc@mail.gmail.com>
	 <5542af940707231709n330ca0ffkc581bb34984b1154@mail.gmail.com>
	 <5542af940707231811t4044fa14rcdc33c9a227a9562@mail.gmail.com>
	 <d3dac270707231857y6dae1f1cq27367b56568301cb@mail.gmail.com>
Message-ID: <d3dac270707231929r558496f6t36dfb40745bb02fe@mail.gmail.com>

Furthermore, since 2^(k+1)*p-1, 2^(k+1)*p+1 must equal -1 and +1
modulo 3, the number 2^(k+1)*p must be 0 modulo 3, implying that p=3.

Therefore, every element of A066466, except 4, must be of the form
3*2^k such that 3*2^(k+1)-1, 3*2^(k+1)+1 are twin primes. There no
such new k (i.e., except known 1,2,6,18) below 1000.

Max

On 7/23/07, Max Alekseyev <maxale at gmail.com> wrote:
> Except element 4, the elements of A066466 have form 2^k*p where p is
> odd prime and both 2^(k+1)*p-1, 2^(k+1)*p+1 are prime (i.e., twin
> primes).
> In other words, A066466 without element 4 is a subsequence of A040040,
> containing elements of the form 2^k*p with prime p.
>
> Max
>
> On 7/23/07, Jonathan Post <jvospost3 at gmail.com> wrote:
> > Any work since 2001 on whether or not there is a 6th anti-prime?  How
> > far has this been searched?  Any proofs or disproofs as to finiteness
> > of A066466?
> >
> > COMMENT FROM Jonathan Vos Post RE A066466
> >
> > %I A066466
> > %S A066466 3, 4, 6, 96, 393216
> > %N A066466 Numbers having just one anti-divisor.
> > %C A066466 Jon Perry calls these anti-prime numbers, saying that these
> > are the only 5 known. This sequence is worth extending, if possible,
> > or proving finite.
> > %F A066466 A066272(a(n)) = 1.
> > %Y A066466 Cf. A066272.
> > %O A066466 1
> > %K A066466 ,more,nonn,
> > %A A066466 Jonathan Vos Post (jvospost2 at yahoo.com), Jul 23 2007
> >
>




From maxale at gmail.com  Tue Jul 24 07:02:50 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Mon, 23 Jul 2007 22:02:50 -0700
Subject: definition of anti-divisor
In-Reply-To: <d3dac270707231929r558496f6t36dfb40745bb02fe@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211317j45f5273fp8aed82da252ff2c8@mail.gmail.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
	 <5542af940707221117t4e411e40o840c9228f2e9b7f8@mail.gmail.com>
	 <3c3af2330707231221y5690ec3cm473f38b5f2bc6adc@mail.gmail.com>
	 <5542af940707231709n330ca0ffkc581bb34984b1154@mail.gmail.com>
	 <5542af940707231811t4044fa14rcdc33c9a227a9562@mail.gmail.com>
	 <d3dac270707231857y6dae1f1cq27367b56568301cb@mail.gmail.com>
	 <d3dac270707231929r558496f6t36dfb40745bb02fe@mail.gmail.com>
Message-ID: <d3dac270707232202q7d6d4198t259627c7b9977368@mail.gmail.com>

On 7/23/07, Max Alekseyev <maxale at gmail.com> wrote:

> Therefore, every element of A066466, except 4, must be of the form
> 3*2^k such that 3*2^(k+1)-1, 3*2^(k+1)+1 are twin primes. There no
> such new k (i.e., except known 1,2,6,18) below 1000.

Small corrections: 1,2,6,18 above correspond to k+1 not k.
In other words, 3*2^n - 1, 3*2^n + 1 are twin primes for n=1,2,6,18.

According to these tables:
http://www.prothsearch.net/riesel.html
http://www.prothsearch.net/riesel2.html
there are no other such n up to 1200000.
Therefore, the next element of A066466 (if it exists) is greater than
3*2^1200000 ~= 10^361236.

Max




From jeremy.gardiner at btinternet.com  Tue Jul 24 11:35:35 2007
From: jeremy.gardiner at btinternet.com (JEREMY GARDINER)
Date: Tue, 24 Jul 2007 10:35:35 +0100 (BST)
Subject: early bird sequence
In-Reply-To: <d3dac270707232202q7d6d4198t259627c7b9977368@mail.gmail.com>
Message-ID: <836624.76115.qm@web86605.mail.ukl.yahoo.com>

Cf.  A048991 and A048992 (Rollman numbers)
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From maximilian.hasler at gmail.com  Tue Jul 24 21:52:51 2007
From: maximilian.hasler at gmail.com (Maximilian Hasler)
Date: Tue, 24 Jul 2007 15:52:51 -0400
Subject: early bird sequence
In-Reply-To: <836624.76115.qm@web86605.mail.ukl.yahoo.com>
References: <d3dac270707232202q7d6d4198t259627c7b9977368@mail.gmail.com>
	 <836624.76115.qm@web86605.mail.ukl.yahoo.com>
Message-ID: <3c3af2330707241252g52e73c0awb1ad943f7562cb7a@mail.gmail.com>

thanks for the reference.

someone could add an explanation to either of those to explain why
they are not the same (e.g. 12 being omitted, 21 is no more "earlier
in the sequence").

M.H.


On 7/24/07, JEREMY GARDINER <jeremy.gardiner at btinternet.com> wrote:
> Cf.  A048991 and A048992 (Rollman numbers)




From maxale at gmail.com  Wed Jul 25 05:30:36 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Tue, 24 Jul 2007 20:30:36 -0700
Subject: stapled intervals (following A090318)
Message-ID: <d3dac270707242030rdf5113wa94c3bd79ed6c689@mail.gmail.com>

SeqFans,

I'm about to submit the following sequences you may find entertaining.

A090318 defines stapled sequence as an interval of positive integers
that does not contain an element coprime to every other element of the
interval. In other word, a sequence is stapled if for every element x
there is another element y (different from x) such that gcd(x,y)>1.

The shortest stapled interval has length 17 and starts with the number 2184.
A090318 gives the smallest stapled interval of the given length n>=17.

In particular, it is interesting to notice that the intervals
[27829,27846] and [27828,27846] are stapled while the interval
[27828,27845] is not.

It is clear that a stapled interval [a,b] may not contain a prime
number greater than b/2 (as such a prime would be coprime to every
other element of the interval). Together with Bertrand's Postulate
that implies a>b/2 or b<2a. And it follows that
* a stapled interval may not contain prime numbers at all;
* for any particular positive integer a, we can determine if it is a
starting point of some stapled interval.

Sequence of starting points of stapled intervals is:
2184, 27828, 27829, 27830, 32214, 57860, 62244, 87890, 92274, 110990,
117920, 122304, 127374, 147950, 151058, 151059, 151060, 151061,
151062, 152334, 163488, 171054, 177980, 182364, 185924, 185925,
185926, 208010, 212394, 238040, 242424, 249678, 260810, 260811,
260812, 260813, 260814, 264498, 268070, 272454, 298100, 302484,
320870, 320871, 320872, 323510, 324564, 328130, 332514, 339434,
339435, 339436, 339437, 339438, 347004, 358160, 362544, 388190,
392574, 399500, 409188, 409189, 409190, 418220, 422600, 422601,
422602, 422603, 422604, 448250, 452634, 471014, 471015, 471016,
478280, 482664

Call a stapled interval "maximum" if it is not a proper sub-interval
of any other stapled interval. Starting points of maximum stapled
intervals are:
2184, 27828, 32214, 57860, 62244, 87890, 92274, 110990, 117920,
122304, 127374, 147950, 151058, 152334, 163488, 171054, 177980,
182364, 185924, 208010, 212394, 238040, 242424, 249678, 260810,
264498, 268070, 272454, 298100, 302484, 320870,  323510, 324564,
328130, 332514, 339434, 347004, 358160, 362544, 388190, 392574,
399500, 409188, 418220, 422600, 448250, 452634, 471014, 478280, 482664

Similarly, call a stapled interval "minimum" if it does not contain
any stapled proper subinterval. Starting points of minimum stapled
intervals are:
2184, 27830, 32214, 57860, 62244, 87890, 92274, 110990, 117920,
122304, 127374, 147950, 151062, 152334, 163488, 171054, 177980,
182364, 185926, 208010, 212394, 238040, 242424, 249678, 260814,
264498, 268070, 272454, 298100, 302484, 320872, 323510, 324564,
328130, 332514, 339438, 347004, 358160, 362544, 388190, 392574,
399500, 409190, 418220, 422604, 448250, 452634, 471016, 478280, 482664

Max




From maxale at gmail.com  Wed Jul 25 05:48:56 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Tue, 24 Jul 2007 20:48:56 -0700
Subject: full rank designs over Z_m
Message-ID: <d3dac270707242048t53cd7745ybdfbbfcee318053f@mail.gmail.com>

SeqFans,

Does the following sounds familiar to anybody?

Fix an integer m>=2. For any positive integer n consider a matrix k x
n over Z_m such that any n rows (out of k) are linearly independent.
Define a(n) as the maximum possible k.

 From practical point of view such construction gives rise to a number
of sequences for different m. Are they in OEIS?

I'm also interested in the theory behind. I believe such matrices
should have been studied.

Thanks,
Max




From bdm at cs.anu.edu.au  Wed Jul 25 06:23:33 2007
From: bdm at cs.anu.edu.au (Brendan McKay)
Date: Wed, 25 Jul 2007 14:23:33 +1000
Subject: full rank designs over Z_m
In-Reply-To: <d3dac270707242048t53cd7745ybdfbbfcee318053f@mail.gmail.com>
References: <d3dac270707242048t53cd7745ybdfbbfcee318053f@mail.gmail.com>
Message-ID: <20070725042333.GA8211@cs.anu.edu.au>


In the case of m not prime, one has to be cautious because several
equivalences from the prime case do not hold.  In particular,
being linearly independent is different from having a full-sized
submatrix (in your case, n*n) of non-zero determinant.

I wrote a paper long ago with Richard Brent that counts integer
matrices with given rank defined by the submatrix-determinant method:
   http://cs.anu.edu.au/~bdm/papers/BrentMcKayZm.pdf
I don't see how it applies to your problem easily, but maybe some
of the ideas are useful.

Also, regardless of your problem, that paper seems to provide any
number of sequences not in OEIS (so this is an invitation for some
enterprising oeiser to jump in).  The paper refers to probabilities
but of course you just need to multiply by the total number of
matrices to turn it into a count.

Brendan.


* Max Alekseyev <maxale at gmail.com> [070725 13:49]:
> SeqFans,
> 
> Does the following sounds familiar to anybody?
> 
> Fix an integer m>=2. For any positive integer n consider a matrix k x
> n over Z_m such that any n rows (out of k) are linearly independent.
> Define a(n) as the maximum possible k.
> 
> >From practical point of view such construction gives rise to a number
> of sequences for different m. Are they in OEIS?
> 
> I'm also interested in the theory behind. I believe such matrices
> should have been studied.
> 
> Thanks,
> Max




From jvospost3 at gmail.com  Wed Jul 25 06:26:38 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Tue, 24 Jul 2007 21:26:38 -0700
Subject: definition of anti-divisor
In-Reply-To: <d3dac270707232202q7d6d4198t259627c7b9977368@mail.gmail.com>
References: <200707211141.l6LBf9dn795643@fry.research.att.com>
	 <d3dac270707211319i7f558c86k2d24383caee3bc59@mail.gmail.com>
	 <5542af940707212354t4b733f7cof101bc0b7960d64b@mail.gmail.com>
	 <5542af940707221117t4e411e40o840c9228f2e9b7f8@mail.gmail.com>
	 <3c3af2330707231221y5690ec3cm473f38b5f2bc6adc@mail.gmail.com>
	 <5542af940707231709n330ca0ffkc581bb34984b1154@mail.gmail.com>
	 <5542af940707231811t4044fa14rcdc33c9a227a9562@mail.gmail.com>
	 <d3dac270707231857y6dae1f1cq27367b56568301cb@mail.gmail.com>
	 <d3dac270707231929r558496f6t36dfb40745bb02fe@mail.gmail.com>
	 <d3dac270707232202q7d6d4198t259627c7b9977368@mail.gmail.com>
Message-ID: <5542af940707242126q68201c55pf5c674ce6848194f@mail.gmail.com>

NEW SEQUENCE FROM Jonathan Vos Post

 %I A000001
 %S A000001 2, 6, 36, 144, 4320, 64800, 777600,
 65318400, 2743372800
 %N A000001 Anti-divisorial; the product of all anti-divisors of all
integers equal or less than n.
 %C A000001 Different from the anti-primorial, which is the partial
products of anti-primes.
 %F A000001 a(n) = PRODUCT[k=3..n] {anti-divisors(k)}
 = PRODUCT[k=3..n] PRODUCT[j=1..A066272(k)] (j-th
 element of k-th row of A130799) = partial products of A091507.
 %e A000001 a(11) = anti-divisors of 3 * anti-divisors of 4 * ... *
anti-divisors of 11 = (2) * (3) * (2 * 3) * (4) * (2 * 3 * 5) * (3 *
5) * (2 * 6) * (3 * 4 * 7) * (2 * 3 * 7) = 2743372800.
 %Y A000001 Cf. A066272, A091507, A130799.
 %O A000001 3,1
 %K A000001 ,easy,nonn,
 %A A000001 Jonathan Vos Post (jvospost2 at yahoo.com), Jul 25 2007
 RH
 RA 192.20.225.32




From maxale at gmail.com  Wed Jul 25 11:49:44 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Wed, 25 Jul 2007 02:49:44 -0700
Subject: ps Re Divisors concatenated shape a prime
In-Reply-To: <33a322bc0707200443l833a591gc4abfcff9e172a43@mail.gmail.com>
References: <200707201002.l6KA2ZKE432256@fry.research.att.com>
	 <d3dac270707200436x7996f230qdeda7d1a243a730e@mail.gmail.com>
	 <33a322bc0707200443l833a591gc4abfcff9e172a43@mail.gmail.com>
Message-ID: <d3dac270707250249u1df96051te33cf39151d8ce1e@mail.gmail.com>

Three more prime divisors have been found with ECM:
1303590319,
2029994921,
and
2635016943923513073981336313737132619.

The co-factor is still composite and requires further factorization.

Regards,
Max

On 7/20/07, Simon Plouffe <simon.plouffe at gmail.com> wrote:
> The number is
> 23637741111132222263396784181836212543250863192525397638\
>     5050794957757619119155152382118123439689236246879378\
>     3543703190673607553698617087406381347215107397221082\
>     2661095832164532219166133479486848572669589736971440\
>     0438460545718008769210914230132785470246166539693602\
>     6557094049233307938690398356410738499619079180796712\
>     8214769992381581114913062399108161968641222982612479\
>     8216323937282334473918719732448590592334050047581378\
>     1681898530966894783743946489718118466810009516275633\
>     6379706181021501427441345045695592720430028548826900\
>     9139118541259851760510992223024564332519703521021984\
>     4460491286637795552815329766690736929975591105630659\
>     5333814738598961996828961134745198615110831211923993\
>     6579222694903972302216624228859904868834042355958453\
>     3249363577198097376680847119169066498726355938826715\
>     6198557234875910075477711877653431239711446975182015\
>     0954106781648014685956717046277302264311087056416726\
>     0822620744350752392673213563296029371913434092554604\
>     5286221741128334521645241488701504785346326116925017\
>     8246786223305225717801965223385003564935724466104514\
>     3560384022108741886504369675409778385289018044217483\
>     7730087393508195567705780212066326225659513109026229\
>     33515586703
>
> and divisible by 13 and 47 if I am not mistaking.
>
> the rest (%/13/47) is not prime either but I can't determine
> its factors.
>
> simon plouffe
>




From zakseidov at yahoo.com  Wed Jul 25 15:42:55 2007
From: zakseidov at yahoo.com (zak seidov)
Date: Wed, 25 Jul 2007 06:42:55 -0700 (PDT)
Subject: A005589: Number of letters in the English name of n, excluding spaces and hyphens
Message-ID: <320334.92018.qm@web38204.mail.mud.yahoo.com>

Dear seqfans,

I'm going to submit 
b005589.txt and a005589.txt
(after end of OEIS vacation?).

In the process, I've found that 
numbers 800 and 900 (with their English names)
are missed in G. Schildberger's file
http://www.research.att.com/~njas/sequences/a000027.txt


If someone's interested I can send both files - 
for checking/extending.  

Thanks, 
Zak

 
A005589  Number of letters in the English name of n,
excluding spaces and hyphens.  
%%%%%%%%%%%

b005589.txt 
0 4
1 3
2 3
3 5
4 4
<skip>
1017 20
1018 20
1019 19
1020 17
1021 20
1022 20
%%%%%%%%%%%%

a005589.txt
zero
one
two
three
four
<skip>
onethousandseventeen
onethousandeightteen
onethousandnineteen
onethousandtwenty
onethousandtwentyone
onethousandtwentytwo


       
____________________________________________________________________________________
Got a little couch potato? 
Check out fun summer activities for kids.
http://search.yahoo.com/search?fr=oni_on_mail&p=summer+activities+for+kids&cs=bz 




I will be doing occasional updates, but that's all.

Neil




From njas at research.att.com  Wed Jul 25 18:51:08 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Wed, 25 Jul 2007 12:51:08 -0400 (EDT)
Subject: The OEIS "vacation" continues through the end of Sept.
In-Reply-To: <320334.92018.qm@web38204.mail.mud.yahoo.com>
References: <320334.92018.qm@web38204.mail.mud.yahoo.com>
Message-ID: <200707251651.l6PGp8YP2000882@fry.research.att.com>

Zak is right.  I'd found that a year ago, when I built a table of the
alphabetical reversal of the standard names of nonnegative integers
from G. Schildberger's draft.

He also had one or two misspellings, as I recall.

On 7/25/07, zak seidov <zakseidov at yahoo.com> wrote:
> Dear seqfans,
>
> I'm going to submit
> b005589.txt and a005589.txt
> (after end of OEIS vacation?).
>
> In the process, I've found that
> numbers 800 and 900 (with their English names)
> are missed in G. Schildberger's file
> http://www.research.att.com/~njas/sequences/a000027.txt
>
>
> If someone's interested I can send both files -
> for checking/extending.
>
> Thanks,
> Zak
>
>
> A005589  Number of letters in the English name of n,
> excluding spaces and hyphens.
> %%%%%%%%%%%
>
> b005589.txt
> 0 4
> 1 3
> 2 3
> 3 5
> 4 4
> <skip>
> 1017 20
> 1018 20
> 1019 19
> 1020 17
> 1021 20
> 1022 20
> %%%%%%%%%%%%
>
> a005589.txt
> zero
> one
> two
> three
> four
> <skip>
> onethousandseventeen
> onethousandeightteen
> onethousandnineteen
> onethousandtwenty
> onethousandtwentyone
> onethousandtwentytwo
>
>
>
> ____________________________________________________________________________________
> Got a little couch potato?
> Check out fun summer activities for kids.
> http://search.yahoo.com/search?fr=oni_on_mail&p=summer+activities+for+kids&cs=bz
>




From petsie at dordos.net  Thu Jul 26 02:36:09 2007
From: petsie at dordos.net (Peter Pein)
Date: Thu, 26 Jul 2007 02:36:09 +0200
Subject: Link to Mathematica script is outdated (says an advertising pest)
Message-ID: <46A7EC79.4000406@dordos.net>

The link to a Mma-script to easy put sequences in the OEIS-format on the
page http://www.research.att.com/~njas/sequences/Submit.html which
should link to a Mma-file http://www.seqfan.net/EISFormat.m leads me to
a strange (hijacking?) site full of advertising which states that
"seqfan.net expired on 07/12/2007 and is pending renewal or deletion."

If I remember correctly, there's a package at mathworld.wolfram.com with
some utilities needed for Eric's project which include some EISFormat
functionality.




From jvospost3 at gmail.com  Thu Jul 26 04:23:36 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Wed, 25 Jul 2007 19:23:36 -0700
Subject: A005589: Number of letters in the English name of n, excluding spaces and hyphens
In-Reply-To: <793412.80456.qm@web38202.mail.mud.yahoo.com>
References: <5542af940707251527x10cfb2c8sefbebd8546c11f59@mail.gmail.com>
	 <793412.80456.qm@web38202.mail.mud.yahoo.com>
Message-ID: <5542af940707251923u5e8e02c4ua885db0f05bceffe@mail.gmail.com>

Zak:

You are right. Same misspellings that I found.

===================

Source: Webster's Revised Unabridged Dictionary (1913)

Eighteen \Eight"een`\, a. [AS. eahtat?ne, eahtat?ne. See
   Eight, and Ten, and cf. Eighty.]
   Eight and ten; as, eighteen pounds.

Eighteen \Eight"een`\, n.
   1. The number greater by a unit than seventeen; eighteen
      units or objects.

   2. A symbol denoting eighteen units, as 18 or xviii.

===================

    * Is Amendment Eighteen Treason? by Joshua Grozier

Author(s) of Review: Charles Hall Davis
Virginia Law Review, Vol. 17, No. 2 (Dec., 1930), pp. 211-214
doi:10.2307/1066265
===================




From petsie at dordos.net  Thu Jul 26 05:19:41 2007
From: petsie at dordos.net (Peter Pein)
Date: Thu, 26 Jul 2007 05:19:41 +0200
Subject: a propos divisors...
In-Reply-To: <46A50268.3040307@dordos.net>
References: <46A50268.3040307@dordos.net>
Message-ID: <46A812CD.60303@dordos.net>

Peter Pein schrieb:
> Dear seqfans,
> 
> I#ve been surprised not to find sequences of the following form in the OEIS:
> 
> a(n)=min(k in N: sigma(r,n)=sigma(r,k)) with sigma(r,n)=sum of the r-th
> power of the divisors of n:
...
> Are these of interest? And if so, up to which exponent r?
> 
> Peter
> 

Although the interst seems to be bounded by a fairly small n (0 until
now), I would like to ask you if you could please help me.

I decided to publish the sequences in two forms:

a) n such there is at least one x<n such that sigma(r,x)=sigma(r,n),
because there exists such a series for r=1 (A069822)

and

b) a(n)=min({k>0: sigma(r,k)=sigma(r,n)})


The issue with these is that they behave in a surprising way for r=3 (I
think, sequences for higher exponents are too hard to calculate without
 tricks)

for r = 0 the type-a-seq starts:
3, 5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20
for r = 1:
11, 15, 17, 23, 25, 26, 31, 35, 38, 39, 41, 46, 47, 51
for r = 2
7, 26, 35, 47, 77, 91, 119, 130, 133, 141, 157, 161, 175

but for r = 3 the air becomes very thin. If I did not make a mistake,
this seq starts(!):

194315, 295301, 590602, 1181204, 1476505, 1886920, 2067107, 2362408,
2526095, 2953010, 3248311, 3691985, 3838913, 4134214, 4469245, 4724816,
5020117, 5610719, 5635135, 5906020

If any of you with some Mathematica knowledge could please confirm, that
the following lines calculate the sequence described above?

Timing[Block[{spa, nmax = 6*10^6, expo = 3},
   Reap[For[n = 1, n <= nmax, n++,
      (If[Head[#1] === spa, #1 = n, Sow[{n, #1}]] & )[
       spa[DivisorSigma[expo, n]]]]][[2,1]]]]

the name spa is an artefact; I tried this with SparseArrays, but the
allowed range of indices has not been sufficient. I use it as an
initially undefined function (Block[{spa..}]). For each n to test I look
wether spa[sigma(r,n)] has been defined. If not, the Head is still spa
and I set spa[sigma(r,n)] to n; else the remembered value together with
n will go to the result via the Sow-Reap mechanism. This way I get
type-a and a hint for type-b sequences in one run of the proggie.

And if you want to make me really happy (I have had birthday on July
24th ;-) (http://www.stevesbeatles.com/songs/when_im_sixty_four.asp this
age will be reached in 20 years, but the symptoms... )):

 If you've got Mathematica and more RAM (4GB or so) than I do (1.5 GB),
could you please run this code  with, say nmax=10 or 20 million? On my
machine it swapped heavily with nmax=6 million and I had to kill
MathKernel as I tried nmax=10^7. The lines above took ~181 seconds to
evaluate (nmax=10^7 has been stopped by me after 15 minutes). I do not
expect any runtimes of more than 7 minutes. Would this be possible, please?

 Alternatively any hints how to calculate these sequences more efficient
would be highly appreciated (AFAIK there exists no kind of "inverse
function" to sigma(r,n) w.r.t. n which could be calculated without this
brute-force method).

Thank you for your attention and in advance for CPU-time,

Peter



* Peter Pein <petsie at dordos.net> [Jul 26. 2007 08:51]:
> The link to a Mma-script to easy put sequences in the OEIS-format on the
> page http://www.research.att.com/~njas/sequences/Submit.html which
> should link to a Mma-file http://www.seqfan.net/EISFormat.m leads me to
> a strange (hijacking?) site full of advertising which states that
> "seqfan.net expired on 07/12/2007 and is pending renewal or deletion."

This is not a domain grabber but "network solutions"

> 
> If I remember correctly, there's a package at mathworld.wolfram.com with
> some utilities needed for Eric's project which include some EISFormat
> functionality.

http://www.aboutus.org/Seqfan.net
ogerard at ext.jussieu.fr (CC)

The domain should simply be renewed.
Else at some point in time a domain grabber will get it.




From arndt at jjj.de  Thu Jul 26 10:50:45 2007
From: arndt at jjj.de (Joerg Arndt)
Date: Thu, 26 Jul 2007 10:50:45 +0200
Subject: Link to Mathematica script is outdated (says an advertising pest)
In-Reply-To: <46A7EC79.4000406@dordos.net>
References: <46A7EC79.4000406@dordos.net>
Message-ID: <20070726085045.GA8464@amd32.purzl.net>

On 6/9/06, Max <maxale at gmail.com> wrote:
> On 6/9/06, Graeme McRae <g_m at mcraefamily.com> wrote:
>
> > It took me quite some time to understand how Ranier Rosenthal is counting
> > rounds, before I obtained A112088 as the number of rounds necessary to kill
> > all but one of n players of the Josephus Game with every third man out.  I
> > first tried using Hugo Pfoertner's idea where the final survivor counts the
> > times he was passed by the executioner including the final rendezvous.  But
> > that method gives an entirely different sequence (A005428:
> > 1,2,3,4,6,9,14,21,31,47,70,105,158,...).
>
> I've also ended up with this exactly sequence in my computations.

I have found an old draft message where I was explaining the details
of my computation.
I reproduce them below in hope that someone will find them useful.

Suppose that we have x persons at the circle (enumerated from 0 to
x-1) and the executioner stands at the person number s. After one
round we will eliminate 1+[(x-s-1)/3] persons and will end up at the
person number (s-x) mod 3. This observation leads to the following
algorithm (stated in the form of PARI/GP program) counting the
required number of rounds:

{ a(n) = local(count = 0, s = 0, x = n, x_new);
while(x>1,
   x_new = x - 1 - (x-s-1)\3;
   s = (s-x) % 3;
   x = x_new;
   count++;
);
count
}

For n=1..50, PARI/GP gives:

? vector(50,n,a(n))
%1 = [0, 1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7,
7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9,
9, 9, 9, 9, 9, 9]

This sequence (counting the number of rounds) seems to be missing in OEIS.

Now we want to "reverse" this algorithm, i.e., starting with a single
person make the given number of rounds r and output n. This involves
solving of the following system of equations:

x_new = x_old - 1 - [(x_old - s_old - 1)/3];
s_new = (s_old - x_old) mod 3;

with respect to x_old and s_old.
It is useful to notice that
[(x_old - s_old - 1)/3] = (x_old - s_old + s_new)/3 - 1,
implying
x_new = x_old - (x_old - s_old + s_new)/3.

Therefore,

x_old = (3*x_new + s_new)/2 - s_old/2.

 From this equation we get value of s_old modulo 2 (it must make x_old
integer). However, that does not uniquely define s_old since it can
take 3 values: 0, 1, and 2. But we can follow greedy strategy here
(though I'm not going to prove that) making x_old the smallest
possible on each round. In other words, we can limit values of s_old
to 1 and 2, and state that

x_old = [(3*x_new + s_new - 1)/2].
s_old = (s_new + x_old) mod 3;

That leads to the following PARI/GP implementation:

{ b(n) = local( x=1, s=0 );
for(r=1,n,
  x = (3*x + s)\2;
  s = (s + x) % 3;
);
x
}

The first 50 values are:

? vector(50,n,b(n))
%1 = [1, 2, 3, 4, 6, 9, 14, 21, 31, 47, 70, 105, 158, 237, 355, 533,
799, 1199, 1798, 2697, 4046, 6069, 9103, 13655, 20482, 30723, 46085,
69127, 103691, 155536, 233304, 349956, 524934, 787401, 1181102,
1771653, 2657479, 3986219, 5979328, 8968992, 13453488, 20180232,
30270348, 45405522, 68108283, 102162425, 153243637, 229865456,
344798184, 517197276]

That is exactly the sequence A005428. Based on our method of
computing, we can give an alternative description of A005428:

Define a mapping f over vectors with two integer components as follows:
f: (x,s)   -->   ( [(3*x + s) / 2], (s + x) mod 3 )

Then A005428(n) is the first component of the n-th iteration of f on
the vector (1,0).

Regards,
Max




From r.rosenthal at web.de  Thu Jul 26 22:24:34 2007
From: r.rosenthal at web.de (Rainer Rosenthal)
Date: Thu, 26 Jul 2007 22:24:34 +0200
Subject: Sequence A112088 Motivation and Example?
In-Reply-To: <d3dac270707261223l69633b85n2277d5adfc2f5869@mail.gmail.com>
References: <4489D93F.3080902@web.de>	 <020401c68c0a$9470a650$6900000a@mcraeclan.com>	 <02a801c68c1c$1421f320$6900000a@mcraeclan.com>	 <d3dac270606092351x3c02aeeo84c20b6871eb7e68@mail.gmail.com> <d3dac270707261223l69633b85n2277d5adfc2f5869@mail.gmail.com>
Message-ID: <46A90302.8020905@web.de>

Max Alekseyev wrote:
> On 6/9/06, Max <maxale at gmail.com> wrote:
>>On 6/9/06, Graeme McRae <g_m at mcraefamily.com> wrote:
>>>It took me quite some time to understand how Ranier Rosenthal is counting
                                                Rainer (please)
>>>rounds, before I obtained A112088 as the number of rounds necessary to kill
>>>all but one of n players of the Josephus Game with every third man out.

Many many thanks for this post!

Some days ago I found my last scribblings, which were meant for
another comment to A112088. But it's all that long ago ... I
didn't understand anything of what I wrote :-(

I'm going to print your mail now and maybe I will find the
energy to get back to all these lovely round-countings.

As far as I remember I wasn't able to really grasp the idea of
the original submitter (Simon Strandgaard). From the discussions
in de.sci.mathematik and de.rec.denksport I remember that there
were some remarks giving hints to Knuth. Oops ... long time ago.

Friendly greetings,
Rainer






From petsie at dordos.net  Fri Jul 27 02:26:33 2007
From: petsie at dordos.net (Peter Pein)
Date: Fri, 27 Jul 2007 02:26:33 +0200
Subject: [Fwd: SEQ FROM Peter Pein]
Message-ID: <46A93BB9.1010209@dordos.net>

Dear Neil,

I obviously made a mistake, when entering the sequence number which I've
got from the dispenser. This should become A131903.

sorry,
Peter

The following is a copy of the email message that was sent to njas
containing the sequence you submitted.

All greater than and less than signs have been replaced by their html
equivalents.  They will be changed back when the message is processed.

This copy is just for your records.  No reply is expected.
 Subject: NEW SEQUENCE FROM Peter Pein


%I A000001
%S A000001 3, 5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22,
23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63
%N A000001 Those integers for which a smaller positive integer exists
which has the same number of divisors
%F A000001 a(n)= n-th element of {x&gt;0, there exists a k with
0&lt;k&lt;x and the same number of divisors as x)
%e A000001 a(4)=8 because it is the fourth integer for which a smaller
integer with the same number of divisors exists (after 3, 5 and 7).
divisors of 8 are 1,2,4,8 which are four and the divisors of 6 which is
less than 8 are (1, 2, 3, 6) which are four.
%t A000001 Clear[tmp];
Function[n, If[Head[#1] === tmp, #1 = n; Unevaluated[Sequence[]], n] &
   [tmp[DivisorSigma[0, n]]]] /@ Range[64]
%Y A000001 Cf. A069822, A131902-A131908
%O A000001 1
%K A000001 ,easy,nonn,
%A A000001 Peter Pein (petsie at dordos.net), Jul 26 2007
RH
RA 192.20.225.32
RU
RI





From petsie at dordos.net  Fri Jul 27 03:38:21 2007
From: petsie at dordos.net (Peter Pein)
Date: Fri, 27 Jul 2007 03:38:21 +0200
Subject: a propos divisors...
In-Reply-To: <46A812CD.60303@dordos.net>
References: <46A50268.3040307@dordos.net> <46A812CD.60303@dordos.net>
Message-ID: <46A94C8D.4090109@dordos.net>

Peter Pein schrieb:
> 
> Although the interst seems to be bounded by a fairly small n (0 until
> now), I would like to ask you if you could please help me.
>
....

> 
> Thank you for your attention and in advance for CPU-time,
> 
> Peter
> 

Well, there are other groups on the net...

THX for ignoring me completely
Peter



Peter,   I at least did not ignore you!   I saved
all your messages to be read later. 

(I just didn't reply yet!)

Neil



>%I A000001
>%S A000001 3, 5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22,
>23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42,
>43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63
>%N A000001 Those integers for which a smaller positive integer exists
>which has the same number of divisors
>%F A000001 a(n)= n-th element of {x&gt;0, there exists a k with
>0&lt;k&lt;x and the same number of divisors as x)
>%e A000001 a(4)=8 because it is the fourth integer for which a smaller
>integer with the same number of divisors exists (after 3, 5 and 7).
>divisors of 8 are 1,2,4,8 which are four and the divisors of 6 which is
>less than 8 are (1, 2, 3, 6) which are four.
>%t A000001 Clear[tmp];
>Function[n, If[Head[#1] === tmp, #1 = n; Unevaluated[Sequence[]], n] &
>   [tmp[DivisorSigma[0, n]]]] /@ Range[64]
>%Y A000001 Cf. A069822, A131902-A131908
>%O A000001 1
>%K A000001 ,easy,nonn,
>%A A000001 Peter Pein (petsie at dordos.net), Jul 26 2007
>RH
>RA 192.20.225.32
>RU
>RI

Appears to be the complement of A007416.

Tony




From njas at research.att.com  Fri Jul 27 04:02:35 2007
From: njas at research.att.com (N. J. A. Sloane)
Date: Thu, 26 Jul 2007 22:02:35 -0400 (EDT)
Subject: a propos divisors...
In-Reply-To: <46A93BB9.1010209@dordos.net>
References: <46A50268.3040307@dordos.net> <46A812CD.60303@dordos.net>
Message-ID: <200707270202.l6R22ZH92491631@fry.research.att.com>

On 7/25/07, Peter Pein <petsie at dordos.net> wrote:

> a) n such there is at least one x<n such that sigma(r,x)=sigma(r,n),
> because there exists such a series for r=1 (A069822)
>
> and
>
> b) a(n)=min({k>0: sigma(r,k)=sigma(r,n)})

[...]

> but for r = 3 the air becomes very thin. If I did not make a mistake,
> this seq starts(!):
>
> 194315, 295301, 590602, 1181204, 1476505, 1886920, 2067107, 2362408,
> 2526095, 2953010, 3248311, 3691985, 3838913, 4134214, 4469245, 4724816,
> 5020117, 5610719, 5635135, 5906020

[...]

> And if you want to make me really happy (I have had birthday on July
> 24th ;-) (http://www.stevesbeatles.com/songs/when_im_sixty_four.asp this
> age will be reached in 20 years, but the symptoms... )):

My congratulations!
Better late than never ;)

>  If you've got Mathematica and more RAM (4GB or so) than I do (1.5 GB),
> could you please run this code  with, say nmax=10 or 20 million? On my
> machine it swapped heavily with nmax=6 million and I had to kill
> MathKernel as I tried nmax=10^7. The lines above took ~181 seconds to
> evaluate (nmax=10^7 has been stopped by me after 15 minutes). I do not
> expect any runtimes of more than 7 minutes. Would this be possible, please?

These are the values below 10^7 that I got with my C++ program using
LiDIA library:

194315 184926
295301 291741
590602 583482
1181204 1166964
1476505 1458705
1886920 1880574
2067107 2042187
2362408 2333928
2526095 2404038
2953010 2917410
3248311 3209151
3691985 3513594
3838913 3792633
4134214 4084374
4469245 4253298
4724816 4667856
5020117 4959597
5610719 5543079
5635135 5362854
5906020 5834820
6023765 5732706
6496622 6418302
6791923 6710043
7382525 7293525
7677826 7585266
7966915 7581966
8268428 8168748
8355545 7951818
8563729 8460489
9132805 8691522
9449632 9335712

Each here line contains a pair:
n k
such that sigma(3,k)=sigma(3,n) and k=a(n) for r=3 (following your
notations above).

I will let my program to run for a couple more days to reach 10^8 bound.

> (AFAIK there exists no kind of "inverse
> function" to sigma(r,n) w.r.t. n which could be calculated without this
> brute-force method).

I disagree with this statement. There is a more or less clever way to
reconstruct the inverse of m=sigma(r,n) w.r.t. n, using integer
factorization of m and a kind of brute-force but of the magnitude of
the number of divisors of m.

Regards,
Max




From petsie at dordos.net  Fri Jul 27 16:32:12 2007
From: petsie at dordos.net (Peter Pein)
Date: Fri, 27 Jul 2007 16:32:12 +0200
Subject: [Fwd: SEQ FROM Peter Pein]
In-Reply-To: <p0624083ec2cf2f4f87a8@[192.168.1.43]>
References: <46A93BB9.1010209@dordos.net> <p0624083ec2cf2f4f87a8@[192.168.1.43]>
Message-ID: <46AA01EC.2070906@dordos.net>

T. D. Noe schrieb:
>> %I A000001
>> %S A000001 3, 5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22,
>> 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42,
>> 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63
>> %N A000001 Those integers for which a smaller positive integer exists
>> which has the same number of divisors
>> %F A000001 a(n)= n-th element of {x&gt;0, there exists a k with
>> 0&lt;k&lt;x and the same number of divisors as x)
>> %e A000001 a(4)=8 because it is the fourth integer for which a smaller
>> integer with the same number of divisors exists (after 3, 5 and 7).
>> divisors of 8 are 1,2,4,8 which are four and the divisors of 6 which is
>> less than 8 are (1, 2, 3, 6) which are four.
>> %t A000001 Clear[tmp];
>> Function[n, If[Head[#1] === tmp, #1 = n; Unevaluated[Sequence[]], n] &
>>   [tmp[DivisorSigma[0, n]]]] /@ Range[64]
>> %Y A000001 Cf. A069822, A131902-A131908
>> %O A000001 1
>> %K A000001 ,easy,nonn,
>> %A A000001 Peter Pein (petsie at dordos.net), Jul 26 2007
>> RH
>> RA 192.20.225.32
>> RU
>> RI
> 
> Appears to be the complement of A007416.
> 
> Tony
> 
Thank you, Tony for this hint.
I ask a bit too late, but:
is it common to let the superseeker have a look at the sequence before
publishing it? I have published seven sequences last night. At the rate
of one request per hour this would last too long.

Peter



* Peter Pein <petsie at dordos.net> [Jul 27. 2007 17:36]:
> [...]

> Thank you, Tony for this hint.
> I ask a bit too late, but:
> is it common to let the superseeker have a look at the sequence before
> publishing it? I have published seven sequences last night. At the rate
> of one request per hour this would last too long.
> 
> Peter

I strongly support to super-seek all new sequences before having them
in the database.

How much resources does one super-seek take?  (CPU & memory)



Peter,  I have put you on the list of "good guys"
who are not subject to the one per hour limit!
Neil




From arndt at jjj.de  Fri Jul 27 17:40:29 2007
From: arndt at jjj.de (Joerg Arndt)
Date: Fri, 27 Jul 2007 17:40:29 +0200
Subject: [Fwd: SEQ FROM Peter Pein]
In-Reply-To: <46AA01EC.2070906@dordos.net>
References: <46A93BB9.1010209@dordos.net> <p0624083ec2cf2f4f87a8@[192.168.1.43]> <46AA01EC.2070906@dordos.net>
Message-ID: <20070727154029.GA22447@amd32.purzl.net>

While preparing to submit to OEIS the decimal digits of two constant
used in the Ramanujan-Lodge Harmonic Number approximation and the
DeTemple-Wang Harmonic Number approximation, I ran into trouble.

I must be doing something really stupid.

Both the low-res Google calculator and the high-res WIMS calculator tell me that
	
((12 * gamma) - 11 - (12 * ln(2))) / (1 - gamma - ln(2^0.5)) = -162.5909

where gamma is Euler's constant.

But Theorem 6 of the arXiv citation, formula (1.13), page 6, insists
that this constant is roughly 1.12150934.

http://arxiv.org/pdf/0707.3950
    Title: Ramanujan's Harmonic Number Expansion into NegativePowers
of a Triangular Number
    Authors: Mark B. Villarino
    Comments: sharp error estimates and general formulas for
Ramanujan's harmonic number expansion
    Subjects: Classical Analysis and ODEs (math.CA); General
Mathematics (math.GM)

    An algebraic transformation of the DeTemple-Wang half-integer
approximation to the harmonic series produces the general formula and
error estimate for the Ramanujan expansion for the nth harmonic number
into negative powers of the nth triangular number. We also discuss the
history of the Ramanujan expansion for the nth harmonic number as well
as sharp estimates of its accuracy, with complete proofs, and we
compare it with other approximative formulas.

As Harminic numbers and Ramanujan appear to popular on EIS, may I ask
for someone to help me in my befuddlement?




From jvospost3 at gmail.com  Fri Jul 27 20:48:04 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Fri, 27 Jul 2007 11:48:04 -0700
Subject: Ramanujan-Lodge's Harmonic Number approximation
In-Reply-To: <563222.67756.qm@web86610.mail.ukl.yahoo.com>
References: <5542af940707270852l3682b86dxdcd8d6dbdabd7ee4@mail.gmail.com>
	 <563222.67756.qm@web86610.mail.ukl.yahoo.com>
Message-ID: <5542af940707271148j2dfc833al93434b370451e97d@mail.gmail.com>

Thank you, Ray and Martin.

I have emailed Mark B. Villarino, using the most recent Costa Rica
email address I could find by Googling (oddly, one is not given in the
arXiv paper), saying, before including your comments (but not giving
out your email addresses nor that of Seqfans):

I enjoyed your paper (and several earlier papers as well on Decartes'
Perfect Lens,  On the Archimedean or Semiregular Polyhedra,  Mertens'
Proof of Mertens' Theorem, ...), and initiated an online conversation
that provides you a correction and a simplification.




From martin_n_fuller at btinternet.com  Fri Jul 27 19:49:50 2007
From: martin_n_fuller at btinternet.com (Martin Fuller)
Date: Fri, 27 Jul 2007 18:49:50 +0100 (BST)
Subject: Ramanujan-Lodge's Harmonic Number approximation
In-Reply-To: <5542af940707270852l3682b86dxdcd8d6dbdabd7ee4@mail.gmail.com>
Message-ID: <563222.67756.qm@web86610.mail.ukl.yahoo.com>

It looks like a typo in the paper.  It should be:
((12 * gamma) - 11 + (12 * ln(2^0.5))) / (1 - gamma - ln(2^0.5))

or more simply:
1 / (1 - gamma - ln(2^0.5)) - 12

The second constant is more simply written as:
1 / (1 - gamma - ln(3/2)) - 54

--- Jonathan Post <jvospost3 at gmail.com> wrote:

> While preparing to submit to OEIS the decimal digits of two constant
> used in the Ramanujan-Lodge Harmonic Number approximation and the
> DeTemple-Wang Harmonic Number approximation, I ran into trouble.
> 
> I must be doing something really stupid.
> 
> Both the low-res Google calculator and the high-res WIMS calculator
> tell me that
> 	
> ((12 * gamma) - 11 - (12 * ln(2))) / (1 - gamma - ln(2^0.5)) =
> -162.5909
> 
> where gamma is Euler's constant.
> 
> But Theorem 6 of the arXiv citation, formula (1.13), page 6, insists
> that this constant is roughly 1.12150934.
> 
> http://arxiv.org/pdf/0707.3950
>     Title: Ramanujan's Harmonic Number Expansion into NegativePowers
> of a Triangular Number
>     Authors: Mark B. Villarino
>     Comments: sharp error estimates and general formulas for
> Ramanujan's harmonic number expansion
>     Subjects: Classical Analysis and ODEs (math.CA); General
> Mathematics (math.GM)
> 
>     An algebraic transformation of the DeTemple-Wang half-integer
> approximation to the harmonic series produces the general formula and
> error estimate for the Ramanujan expansion for the nth harmonic
> number
> into negative powers of the nth triangular number. We also discuss
> the
> history of the Ramanujan expansion for the nth harmonic number as
> well
> as sharp estimates of its accuracy, with complete proofs, and we
> compare it with other approximative formulas.
> 
> As Harminic numbers and Ramanujan appear to popular on EIS, may I ask
> for someone to help me in my befuddlement?
> 




I'm curious how a sieve, similar to the sieve of Erosthanese, would
"perform".  The basic idea is this: suppose that n is some number that you
want to factor. Assume that you can map, for x = 0 to some upper bound less
than the smallest factor of n, n == a mod x --> f == b mod x, where f is a
factor of n. Say, for instance:

n == 2 mod 3, therefore f == 1 mod 3
n == 1 mod 4, therefore f == 3 mod 4
n == 4 mod 5, therefore f == 2 mod 5

& etc.

Thus we could establish a "profile" for f, which can be turned into a sieve
by finding all integers congruent to 1 mod 3, then a subset congruent to 3
mod 4, then a subset of that congruent to 2 mod 5, etc. My question is how
"quickly" can we narrow in on possible values for f by using this sieve? I
think something like this may be possible for values of n with certain
properties, but I'm not sure how well it'd perform.







From aplewe at sbcglobal.net  Fri Jul 27 21:28:30 2007
From: aplewe at sbcglobal.net (Andrew Plewe)
Date: Fri, 27 Jul 2007 12:28:30 -0700
Subject: A theoretical question -- sieving via n mod x
In-Reply-To: <5542af940707271148j2dfc833al93434b370451e97d@mail.gmail.com>
References: <5542af940707270852l3682b86dxdcd8d6dbdabd7ee4@mail.gmail.com>
	 <563222.67756.qm@web86610.mail.ukl.yahoo.com>
	 <5542af940707271148j2dfc833al93434b370451e97d@mail.gmail.com>
Message-ID: <200707271935.l6RJZCci090307@shiva.jussieu.fr>

I just submitted 4 related sequences, A131915 through A131918, which
are the digits (1st 100) of the decimal expansions of, and the
continued fraction expansions (100 integers) of the two formulae, in
all cases crediting Martin Fuller's simplifications and/or corrections
of Mark B. Villarino's paper.




From jvospost3 at gmail.com  Sat Jul 28 02:38:51 2007
From: jvospost3 at gmail.com (Jonathan Post)
Date: Fri, 27 Jul 2007 17:38:51 -0700
Subject: Ramanujan-Lodge's Harmonic Number approximation
In-Reply-To: <5542af940707271735t4ea8f890y39802dd9c657193@mail.gmail.com>
References: <5542af940707270852l3682b86dxdcd8d6dbdabd7ee4@mail.gmail.com>
	 <563222.67756.qm@web86610.mail.ukl.yahoo.com>
	 <5542af940707271148j2dfc833al93434b370451e97d@mail.gmail.com>
	 <5542af940707271735t4ea8f890y39802dd9c657193@mail.gmail.com>
Message-ID: <5542af940707271738m4533a2e7rbe837493872723c8@mail.gmail.com>

Synchronicity strikes!  At the same minute, I received this email from
the author of the arXiv paper which Ray Chandler helped me determine
had a typo, and Martin Fuller corrected and simplified the formulae:

==============

Dear Jonathan Vos Post,

Thank you so very much for your letter.  You are quite right, and I
have incorporated the correction/simplification into my preprint.  It
should appear on lanl.xxx on Monday.

I am very grateful for the trouble you took and, as a naturally
prejudiced author, I thank you for your kind words about my other preprints.

The paper on semiregular polyhedra will appear in Elemente der
Mathematik...it has been accepted for publication as part of the
"Euler year."

The paper on "The Probability of a Run" was published in the March,
2007,  issue of the Mathematical Gazette.

Finally, the paper on the accuracy or Ramanujan's approximation to the
arc length of an ellipse was published in the online journal JIPAM
(Journal of Inequalities in Pure and Applied Mathematics) in January
of 2006.

I would be delighted to join into any further dialogue about the paper.

Again, thank you very much.

Sincerely yours,
MARK B. VILLARINO

MARK B. VILLARINO
Escuela de matem?tica
Universidad de Costa Rica
San Jos?, Costa RicaTelephone: 506-857-4679
email: mvillari at cariari.ucr.ac.cr
email: mark.villarino at gmail.com

==============





From franktaw at netscape.net  Sat Jul 28 02:43:56 2007
From: franktaw at netscape.net (franktaw at netscape.net)
Date: Fri, 27 Jul 2007 20:43:56 -0400
Subject: A theoretical question -- sieving via n mod x
In-Reply-To: <200707271935.l6RJZCci090307@shiva.jussieu.fr>
References: <200707271935.l6RJZCci090307@shiva.jussieu.fr>
Message-ID: <8C99EC783AE068B-5F0-7232@mblk-d13.sysops.aol.com>

Assuming that evaluating your function is fast, this should be very 
efficient.  Combining results in different moduli is essentially just 
Euclid's algorithm (for the GCD) -- which is quite fast -- followed by 
a couple of multiplications and an addition.

Franklin T. Adams-Watters

-----Original Message-----
From: Andrew Plewe <aplewe at sbcglobal.net>

I'm curious how a sieve, similar to the sieve of Erosthanese, would
"perform".  The basic idea is this: suppose that n is some number that 
you
want to factor. Assume that you can map, for x = 0 to some upper bound 
less
than the smallest factor of n, n == a mod x --> f == b mod x, where f 
is a
factor of n. Say, for instance:

n == 2 mod 3, therefore f == 1 mod 3
n == 1 mod 4, therefore f == 3 mod 4
n == 4 mod 5, therefore f == 2 mod 5

& etc.

Thus we could establish a "profile" for f, which can be turned into a 
sieve
by finding all integers congruent to 1 mod 3, then a subset congruent 
to 3
mod 4, then a subset of that congruent to 2 mod 5, etc. My question is 
how
"quickly" can we narrow in on possible values for f by using this 
sieve? I
think something like this may be possible for values of n with certain
properties, but I'm not sure how well it'd perform.

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Check Out the new free AIM(R) Mail -- Unlimited storage and 
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There are three seqs titled "Number of weighted voting procedures"
http://www.research.att.com/~njas/sequences/A005254
http://www.research.att.com/~njas/sequences/A005256
http://www.research.att.com/~njas/sequences/A005257

Could somebody define the term "weighted voting procedure"
and add the information to differentiate the seqs?




Hello seqfans,

Here I see two entries with the same name:

A030225 Number of n-celled polyhexes (hexagonal polyominoes) with bilateral symmetry.
A002215 Number of polyhexes with n hexagons, having reflectional symmetry. 


Probably, the second one should be about "restricted" polyhexes, BUT, there are two entries about restricted polyhexes:

A002216 Harary-Read numbers: restricted hexagonal polyominoes (cata-polyhexes) with n cells. 
A002212 Number of restricted hexagonal polyominoes with n cells. 


On top of that, if a2216 is the number of cata-polyhexes, then what is
A038142 Number of planar cata-polyhexes with n cells.

I am completely confused.

Best, Tanya


_________________________________________________________________
Need personalized email and website? Look no further. It's easy
with Doteasy $0 Web Hosting! Learn more at www.doteasy.com



You did such a great job removing duplicate sequences. How about duplicate definitions:

A002212 Number of restricted hexagonal polyominoes with n cells.
A005963 Number of restricted hexagonal polyominoes with n cells. 

Tanya


_________________________________________________________________
Need personalized email and website? Look no further. It's easy
with Doteasy $0 Web Hosting! Learn more at www.doteasy.com



A001168 Number of fixed polyominoes with n cells.
A006762 Number of fixed polyominoes with n cells. 


_________________________________________________________________
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with Doteasy $0 Web Hosting! Learn more at www.doteasy.com




Let me remind folks that the primary purpose
of the OEIS is to give pointers to the literature,
you can find out who else has studied the same sequence.

I don't claim to give precise definitions for every single

If you find two sequences with the same definition
("Related to the enumeration of ***", say),
feel free to track down the references and

Neil




From arndt at jjj.de  Sat Jul 28 12:06:39 2007
From: arndt at jjj.de (Joerg Arndt)
Date: Sat, 28 Jul 2007 12:06:39 +0200
Subject: weighted voting sequences
Message-ID: <20070728100639.GA15382@amd32.purzl.net>

so that when you come across a sequence in your work, 
sequence - that would take more time than I have available.
send me more precise definitions.
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Message-ID: <5542af940707281505n357500fbu2fada9399c6b518c at mail.gmail.com>
Date: Sat, 28 Jul 2007 15:05:59 -0700
From: "Jonathan Post" <jvospost3 at gmail.com>
To: "Max Alekseyev" <maxale at gmail.com>
Subject: Re: definition of anti-divisor
Cc: "Maximilian Hasler" <maximilian.hasler at gmail.com>,
   "Sequence Fans" <seqfan at ext.jussieu.fr>, jvospost2 at yahoo.com
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	 <5542af940707221117t4e411e40o840c9228f2e9b7f8 at mail.gmail.com>
	 <3c3af2330707231221y5690ec3cm473f38b5f2bc6adc at mail.gmail.com>
	 <5542af940707231709n330ca0ffkc581bb34984b1154 at mail.gmail.com>
	 <5542af940707231811t4044fa14rcdc33c9a227a9562 at mail.gmail.com>
	 <d3dac270707231857y6dae1f1cq27367b56568301cb at mail.gmail.com>
	 <d3dac270707231929r558496f6t36dfb40745bb02fe at mail.gmail.com>
	 <d3dac270707232202q7d6d4198t259627c7b9977368 at mail.gmail.com>
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NEW SEQUENCE FROM Jonathan Vos Post

%I A000001
%S A000001 3, 18, 1728, 679477248
%N A000001 Anti-primorials, partial products of anti-primes A092680.
%C A000001 This is to primorial (A002110) as anti-prime (A092680) is
to prime (A000040). Max Alekseyev <maxale at gmail.com> points out
 that every element of A066466, except 4, must be of the form 3*2^k
such that 3*2^(k+1)-1, 3*2^(k+1)+1 are twin primes. There no such new
k+1 (i.e., except known 1,2,6,18) below 1000.
In other words, 3*2^n - 1, 3*2^n + 1 are twin primes for n=1,2,6,18.
 According to these tables:
http://www.prothsearch.net/riesel.html
http://www.prothsearch.net/riesel2.html
there are no other such n up to 1200000.
Therefore, the next element of A066466 (if it exists) is greater than
3*2^1200000 ~= 10^361236. Hence the next element of the
anti-primorials (if it exists) is greater than 679477248 *3*2^1200000
~= 679477248 *
 10^361236 ~= 6 * 10^361245.
%D A000001 1.
%F A000001 a(n) = PRODUCT[k = 1..n] A092680(k).
%e A000001 a(1) = 3.
a(2) = 3 * 6 = 18.
a(3) = 3 * 6 * 96 = 1728.
a(4) = 3 * 6 * 96 * 393216 = 679477248.
%Y A000001 Cf. A000040, A002110, A092680, A130874.
%O A000001 1,1
%K A000001 ,nonn,unkn,
%A A000001 Jonathan Vos Post (jvospost2 at yahoo.com), Jul 28 2007
RH
RA 192.20.225.32




From petsie at dordos.net  Sun Jul 29 01:04:43 2007
From: petsie at dordos.net (Peter Pein)
Date: Sun, 29 Jul 2007 01:04:43 +0200
Subject: a propos divisors...
In-Reply-To: <d3dac270707270520r26d2f151rf531267baad94f02@mail.gmail.com>
References: <46A50268.3040307@dordos.net> <46A812CD.60303@dordos.net> <d3dac270707270520r26d2f151rf531267baad94f02@mail.gmail.com>
Message-ID: <46ABCB8B.2090809@dordos.net>

Max Alekseyev schrieb:
> On 7/25/07, Peter Pein <petsie at dordos.net> wrote:
> 
>> a) n such there is at least one x<n such that sigma(r,x)=sigma(r,n),
>> because there exists such a series for r=1 (A069822)
>>
>> and
>>
>> b) a(n)=min({k>0: sigma(r,k)=sigma(r,n)})
> 
> [...]
> 
>> but for r = 3 the air becomes very thin. If I did not make a mistake,
>> this seq starts(!):
>>
>> 194315, 295301, 590602, 1181204, 1476505, 1886920, 2067107, 2362408,
>> 2526095, 2953010, 3248311, 3691985, 3838913, 4134214, 4469245, 4724816,
>> 5020117, 5610719, 5635135, 5906020
> 
> [...]
> 
>> And if you want to make me really happy (I have had birthday on July
>> 24th ;-) (http://www.stevesbeatles.com/songs/when_im_sixty_four.asp this
>> age will be reached in 20 years, but the symptoms... )):
> 
> My congratulations!
> Better late than never ;)

Thank you!

> 
>>  If you've got Mathematica and more RAM (4GB or so) than I do (1.5 GB),
>> could you please run this code  with, say nmax=10 or 20 million? On my
>> machine it swapped heavily with nmax=6 million and I had to kill
>> MathKernel as I tried nmax=10^7. The lines above took ~181 seconds to
>> evaluate (nmax=10^7 has been stopped by me after 15 minutes). I do not
>> expect any runtimes of more than 7 minutes. Would this be possible, please?
> 
> These are the values below 10^7 that I got with my C++ program using
> LiDIA library:
> 
> 194315 184926
> 295301 291741
> 590602 583482
> 1181204 1166964
> 1476505 1458705
> 1886920 1880574
> 2067107 2042187
> 2362408 2333928
> 2526095 2404038
> 2953010 2917410
> 3248311 3209151
> 3691985 3513594
> 3838913 3792633
> 4134214 4084374
> 4469245 4253298
> 4724816 4667856
> 5020117 4959597
> 5610719 5543079
> 5635135 5362854
> 5906020 5834820
> 6023765 5732706
> 6496622 6418302
> 6791923 6710043
> 7382525 7293525
> 7677826 7585266
> 7966915 7581966
> 8268428 8168748
> 8355545 7951818
> 8563729 8460489
> 9132805 8691522
> 9449632 9335712
> 
> Each here line contains a pair:
> n k
> such that sigma(3,k)=sigma(3,n) and k=a(n) for r=3 (following your
> notations above).
> 
> I will let my program to run for a couple more days to reach 10^8 bound.
> 
>> (AFAIK there exists no kind of "inverse
>> function" to sigma(r,n) w.r.t. n which could be calculated without this
>> brute-force method).
> 
> I disagree with this statement. There is a more or less clever way to
> reconstruct the inverse of m=sigma(r,n) w.r.t. n, using integer
> factorization of m and a kind of brute-force but of the magnitude of
> the number of divisors of m.
> 
> Regards,
> Max
> 
Dear Max,

:-) thank you very much! :-)

I just entered the extension to A13190{7|8} and of course mentioned your
name. When your program gives more results, please extend the sequences
or if you've got a lot of numbers, send Neil a b-file.

The method you mention sounds mangeable - I will look for that algorithm.

Thanks again,
Peter



Dear seqfans,
I'd appreciate information (including references) 
about conditions (necessary and/or sufficient) on 
a sequence of positive numbers a1, a2, a3, ... 
in order that their Hankel matrix be positive 
definite. 
Thanks,
Emeric



Dear seqfans,
I'd appreciate information (including references) 
about conditions (necessary and/or sufficient) on 
a sequence of positive numbers a1, a2, a3, ... 
in order that their Hankel matrix be positive 
definite. 
Thanks,
Emeric




From deutsch at duke.poly.edu  Sun Jul 29 06:12:09 2007
From: deutsch at duke.poly.edu (deutsch)
Date: Sun, 29 Jul 2007 04:12:09 GMT
Subject: positive definiteness of Hankel matrices
In-Reply-To: <200707280825.AA3347841186@TanyaKhovanova.com>
References: <200707280825.AA3347841186@TanyaKhovanova.com>
Message-ID: <46ac1399.275.718.7202@duke.poly.edu>

A006762 is counting only polyominoes that are strictly 2-dimensional - 
it excludes those where all the squares are in a single line.  Thus, 
for n>1, A006762(n) = A001168(n) - 2.

Franklin T. Adams-Watters

-----Original Message-----
From: Tanya Khovanova <tanyakh at TanyaKhovanova.com>

A001168 Number of fixed polyominoes with n cells.
A006762 Number of fixed polyominoes with n cells.

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From neoneye at gmail.com  Sun Jul 29 19:13:08 2007
From: neoneye at gmail.com (Simon Strandgaard)
Date: Sun, 29 Jul 2007 19:13:08 +0200
Subject: Sequence A112088 Motivation and Example?
In-Reply-To: <46A90302.8020905@web.de>
References: <4489D93F.3080902@web.de>
	 <020401c68c0a$9470a650$6900000a@mcraeclan.com>
	 <02a801c68c1c$1421f320$6900000a@mcraeclan.com>
	 <d3dac270606092351x3c02aeeo84c20b6871eb7e68@mail.gmail.com>
	 <d3dac270707261223l69633b85n2277d5adfc2f5869@mail.gmail.com>
	 <46A90302.8020905@web.de>
Message-ID: <df1390cc0707291013q696575d9hd69b17382edb8141@mail.gmail.com>

On 7/26/07, Rainer Rosenthal <r.rosenthal at web.de> wrote:
> Max Alekseyev wrote:
> > On 6/9/06, Max <maxale at gmail.com> wrote:
> >>On 6/9/06, Graeme McRae <g_m at mcraefamily.com> wrote:
> >>>It took me quite some time to understand how Ranier Rosenthal is counting
>                                                 Rainer (please)
> >>>rounds, before I obtained A112088 as the number of rounds necessary to kill
> >>>all but one of n players of the Josephus Game with every third man out.
>
> Many many thanks for this post!
>
> Some days ago I found my last scribblings, which were meant for
> another comment to A112088. But it's all that long ago ... I
> didn't understand anything of what I wrote :-(
>
> I'm going to print your mail now and maybe I will find the
> energy to get back to all these lovely round-countings.
>
> As far as I remember I wasn't able to really grasp the idea of
> the original submitter (Simon Strandgaard). From the discussions
> in de.sci.mathematik and de.rec.denksport I remember that there
> were some remarks giving hints to Knuth. Oops ... long time ago.

I wrote a step-by-step guide to how a112088 works, see
http://www.research.att.com/~njas/sequences/a112088.html

maybe it can be useful to you?

-- 
Simon Strandgaard




From Eric.Angelini at kntv.be  Mon Jul 30 00:11:35 2007
From: Eric.Angelini at kntv.be (Eric Angelini)
Date: Mon, 30 Jul 2007 00:11:35 +0200
Subject: Substrings in A046043 
Message-ID: <F05775148D000C4B9321A5113D53CB7D50D245@KNTV-SERVER01.kntv.local>

Hello SeqFans,

Integers of A046043 are interpreted like this:

0123
1210 = one 0 two 1 one 2 zero 3 -> one,two,one,zero = 1210

0123
2020 = two 0 zero 1 two 2 zero 3 -> two,zero,two,zero = 2020

01234
21200 = two 0 one 1 two 2 zero 3 zero 4 -> two,one,two,zero,zero = 21200

0123456
3211000 = three 0 two 1 one 2 one 3 zero 4 zero 5 zero 6 -> etc. = 3211000 

...

0123456789
6210001000 = six 0 two 1 one 2 zero 3 zero 4 zero 5 one 6 zero 7 zero 8 zero 9

Now if we add an extra string "10" in the upper line we could perhaps go on in
the sequence like this:

012345678910
53110100002

... meaning that we have in the integer 53110100002:

- five (sub)strings "0"
- three strings "1"
- one string  2
- one string  3
- zero string 4
- one string  5
- zero string 6
- zero string 7
- zero string 8
- zero string 9
- two strings 10

What would be the next integers in the sequence?

Best,
E.

 


 

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From Eric.Angelini at kntv.be  Mon Jul 30 00:45:48 2007
From: Eric.Angelini at kntv.be (Eric Angelini)
Date: Mon, 30 Jul 2007 00:45:48 +0200
Subject: =?iso-8859-1?Q?RE=A0=3A_Substrings_in_A046043_?=
Message-ID: <F05775148D000C4B9321A5113D53CB7D50D246@KNTV-SERVER01.kntv.local>

...

> meaning that we have in the integer 53110100002 (...)

... The length of such an integer indicates obviously how many
(sub)strings have to be discribed:
53110100002 has 11 digits thus the 11 substrings "0", "1"... to "10"
must be described

> What would be the next integers in the sequence?

... I don't know if there is something before 62200010001:

012345678910
62200010001 

Best,
E.

 


 

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From qq-quet at mindspring.com  Mon Jul 30 16:23:05 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Mon, 30 Jul 07 08:23:05 -0600
Subject: Do any integers occur in both sequences?
In-Reply-To: <46ABCB8B.2090809@dordos.net>
References: <46A50268.3040307@dordos.net> <46A812CD.60303@dordos.net>
	 <d3dac270707270520r26d2f151rf531267baad94f02@mail.gmail.com>
	 <46ABCB8B.2090809@dordos.net>
Message-ID: <E1IFWBz-0000Pz-00@pop03.mail.atl.earthlink.net>

------=_Part_17373_17766144.1185831725052
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 7bit
Content-Disposition: inline

On 7/28/07, Peter Pein <petsie at dordos.net> wrote:

> > Each here line contains a pair:
> > n k
> > such that sigma(3,k)=sigma(3,n) and k=a(n) for r=3 (following your
> > notations above).
> >
> > I will let my program to run for a couple more days to reach 10^8 bound.

I stopped the program after it reached 7*10^7 as it started to eat too
much memory.
The results are attached. Please format them and submit to OEIS as you
feel appropriate.

P.S. Eventually I've defended Ph.D. in Computer Science ;)

Regards,
Max

------=_Part_17373_17766144.1185831725052
Content-Type: text/plain; name=sigma3.txt; charset=ANSI_X3.4-1968
Content-Transfer-Encoding: base64
X-Attachment-Id: f_f4rhifo0
Content-Disposition: attachment; filename="sigma3.txt"

MTk0MzE1IDE4NDkyNgoyOTUzMDEgMjkxNzQxCjU5MDYwMiA1ODM0ODIKMTE4
MTIwNCAxMTY2OTY0CjE0NzY1MDUgMTQ1ODcwNQoxODg2OTIwIDE4ODA1NzQK
MjA2NzEwNyAyMDQyMTg3CjIzNjI0MDggMjMzMzkyOAoyNTI2MDk1IDI0MDQw
MzgKMjk1MzAxMCAyOTE3NDEwCjMyNDgzMTEgMzIwOTE1MQozNjkxOTg1IDM1
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OQo=

------=_Part_17373_17766144.1185831725052--




From petsie at dordos.net  Tue Jul 31 04:32:57 2007
From: petsie at dordos.net (Peter Pein)
Date: Tue, 31 Jul 2007 04:32:57 +0200
Subject: Do any integers occur in both sequences?
In-Reply-To: <E1IFWBz-0000Pz-00@pop03.mail.atl.earthlink.net>
References: <E1IFWBz-0000Pz-00@pop03.mail.atl.earthlink.net>
Message-ID: <46AE9F59.9010203@dordos.net>

Leroy Quet schrieb:
> I have just submitted these two interdependent sequences (So don't look 
> for them in the database yet):
> 
>> %I A131937
>> %S A131937 1,4,8,14,21,29,38,49,61
>> %N A131937 a(1)=1; a(2)=4. a(n) = a(n-1) + (nth positive integer which 
>> does not occur in sequence A131938).
>> %e A131937 A131938: 2,5,10,16,23,32,42,53,...
>> Positive integers not in A131938: 1,3,4,6,7,8,9,11,...
>> So A131937(8) = A131937(7) + 11 = 49.
>> %Y A131937 A131938
>> %O A131937 1
>> %K A131937 ,more,nonn,
> 
>> %I A131938
>> %S A131938 2,5,10,16,23,32,42,53,65,78,93,109
>> %N A131938 a(1)=2; a(2)=5. a(n) = a(n-1) + (nth positive integer which 
>> does not occur in sequence A131937).
>> %e A131938 A131937: 1,4,8,14,21,29,...
>> Positive integers not in A131937: 2,3,5,6,7,9,10,11,...
>> So A131938(8) = A131938(7) + 11 = 53.
>> %Y A131938 A131937
>> %O A131938 1
>> %K A131938 ,more,nonn,
> 
> 
> I have not thought about this too hard; so for all I know, the proof is 
> quite easy.
> 
> Do any positive integers occur in both A131937 and A131938?
> 
> Thanks,
> Leroy Quet
> 

Hello Leroy,

let's just do the first step (n=3) to extend these lists:
(let A131937=:l1 and A131938=:l2)

I get
Complement[posint, l2]={1,3,4,6,7,8,9,...}
Therefore the third positive integer which is not in l2 is 4 again and
l1 starts {1, 4, 4, 6, 8, 10, 12}
and l2 begins {2, 5, 5, 7, 9, 11, 13}

The Mma-Code and its output:

each "grouped output" consists of l1 & l2 so far, n followed by the
complement of (the begin of) N w.r.t. l2 and l1 resp.; you can count the
elements using the latter.

In[26]:=
list1={1,4};list2={2,5};n=3;
Do[
    Print[{list1,list2}];
    Print[n," ",Complement[Range[15],#]&/@{list2,list1}];
    AppendTo[list1,Part[Complement[Range[n+Length[list2]],list2],n]];
    AppendTo[list2,Part[Complement[Range[n+Length[list1]],list1],n]];
    n++;
    ,{5}];
list1
list2
Intersection[%%,%]

 From In[26]:=
{{1,4},{2,5}}
 From In[26]:=
3 {{1,3,4,6,7,8,9,10,11,12,13,14,15},
   {2,3,5,6,7,8,9,10,11,12,13,14,15}}
 From In[26]:=
{{1,4,4},{2,5,5}}
 From In[26]:=
4 {{1,3,4,6,7,8,9,10,11,12,13,14,15},
   {2,3,5,6,7,8,9,10,11,12,13,14,15}}
 From In[26]:=
{{1,4,4,6},{2,5,5,7}}
 From In[26]:=
5 {{1,3,4,6,8,9,10,11,12,13,14,15},
   {2,3,5,7,8,9,10,11,12,13,14,15}}
 From In[26]:=
{{1,4,4,6,8},{2,5,5,7,9}}
 From In[26]:=
6 {{1,3,4,6,8,10,11,12,13,14,15},
   {2,3,5,7,9,10,11,12,13,14,15}}
 From In[26]:=
{{1,4,4,6,8,10},{2,5,5,7,9,11}}
 From In[26]:=
7 {{1,3,4,6,8,10,12,13,14,15},
   {2,3,5,7,9,11,12,13,14,15}}
Out[28]= (* A131937 *)
{1,4,4,6,8,10,12}
Out[29]= (* A131938 *)
{2,5,5,7,9,11,13}
Out[30]= (* intersection *)
{}

These are the first elements of l1 and l2 after 100 iterations:

{1,4,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,\
54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,\
104,106,108,110,112,114,116,118,120,122,124,126,128,130,132,134,136,138,140,\
142,144,146,148,150,152,154,156,158,160,162,164,166,168,170,172,174,176,178,\
180,182,184,186,188,190,192,194,196,198,200,202}

{2,5,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,\
55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,97,99,101,103,\
105,107,109,111,113,115,117,119,121,123,125,127,129,131,133,135,137,139,141,\
143,145,147,149,151,153,155,157,159,161,163,165,167,169,171,173,175,177,179,\
181,183,185,187,189,191,193,195,197,199,201,203}

and the intersection is empty.

Looks like the algorithm separates odd and even numbers.

Because the algorithm is very inefficient, I've chosen 10^4 as maximum
number of iterations:

list1={1,4};list2={2,5};n=3;
Do[
    AppendTo[list1,Part[Complement[Range[n+Length[list2]],list2],n]];
    AppendTo[list2,Part[Complement[Range[n+Length[list1]],list1],n]];
    n++;
    ,{10^4}];
Intersection[list1,list2]

--> {}

and the lists (almost) are the evens and the odds:

Take[#,-5]&/@{list1,list2}
{{19994,19996,19998,20000,20002},{19995,19997,19999,20001,20003}}

I guess, something went wrong in your definition or in my algorithm.


Best regards,
Peter




From petsie at dordos.net  Tue Jul 31 05:11:58 2007
From: petsie at dordos.net (Peter Pein)
Date: Tue, 31 Jul 2007 05:11:58 +0200
Subject: Do any integers occur in both sequences?
In-Reply-To: <E1IFWBz-0000Pz-00@pop03.mail.atl.earthlink.net>
References: <E1IFWBz-0000Pz-00@pop03.mail.atl.earthlink.net>
Message-ID: <46AEA87E.1010709@dordos.net>

Leroy Quet schrieb:
> I have just submitted these two interdependent sequences (So don't look 
> for them in the database yet):
> 
>> %I A131937
>> %S A131937 1,4,8,14,21,29,38,49,61
>> %N A131937 a(1)=1; a(2)=4. a(n) = a(n-1) + (nth positive integer which 
>> does not occur in sequence A131938).
>> %e A131937 A131938: 2,5,10,16,23,32,42,53,...
>> Positive integers not in A131938: 1,3,4,6,7,8,9,11,...
>> So A131937(8) = A131937(7) + 11 = 49.
>> %Y A131937 A131938
>> %O A131937 1
>> %K A131937 ,more,nonn,
> 
>> %I A131938
>> %S A131938 2,5,10,16,23,32,42,53,65,78,93,109
>> %N A131938 a(1)=2; a(2)=5. a(n) = a(n-1) + (nth positive integer which 
>> does not occur in sequence A131937).
>> %e A131938 A131937: 1,4,8,14,21,29,...
>> Positive integers not in A131937: 2,3,5,6,7,9,10,11,...
>> So A131938(8) = A131938(7) + 11 = 53.
>> %Y A131938 A131937
>> %O A131938 1
>> %K A131938 ,more,nonn,
> 
> 
> I have not thought about this too hard; so for all I know, the proof is 
> quite easy.
> 
> Do any positive integers occur in both A131937 and A131938?
> 
> Thanks,
> Leroy Quet
> 
Hi again,

even if you meant "n-th pos. int. which does yet not occur in either l1
or l2" (or does one say "..does yet neither occur in l1 nor in l2"?) I
get another result

list1={1,4};list2={2,5};n=3;
Do[
    Print[{list1,list2}];
    Print["n= ",n," ",Complement[Range[25],list2]];
    AppendTo[list1,Part[Complement[Range[3n ],Union[list1,list2]],n]];
    Print["n= ",n," ",Complement[Range[25],list1]];
    AppendTo[list2,Part[Complement[Range[3n],Union[list1,list2]],n]];
    n++;
    ,{5}];
list1
list2
Intersection[%%,%]


{{1,4},{2,5}}
n= 3 {1,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
n= 3 {2,3,5,6,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
{{1,4,7},{2,5,8}}
n= 4 {1,3,4,6,7,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
n= 4 {2,3,5,6,8,9,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
{{1,4,7,10},{2,5,8,11}}
n= 5 {1,3,4,6,7,9,10,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
n= 5 {2,3,5,6,8,9,11,12,14,15,16,17,18,19,20,21,22,23,24,25}
{{1,4,7,10,13},{2,5,8,11,14}}
n= 6 {1,3,4,6,7,9,10,12,13,15,16,17,18,19,20,21,22,23,24,25}
n= 6 {2,3,5,6,8,9,11,12,14,15,17,18,19,20,21,22,23,24,25}
{{1,4,7,10,13,16},{2,5,8,11,14,17}}
n= 7 {1,3,4,6,7,9,10,12,13,15,16,18,19,20,21,22,23,24,25}
n= 7 {2,3,5,6,8,9,11,12,14,15,17,18,20,21,22,23,24,25}

l1: {1,4,7,10,13,16,19}
l2: {2,5,8,11,14,17,20}
intersection: {}

And l1, l2 and intersection for the first 100 iterations are (sorry for
weird linewidths):

{1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52,
55, 58, 61, 64, 67, 70,
  73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118,
121, 124, 127, 130,
  133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172,
175, 178, 181, 184,
  187, 190, 193, 196, 199, 202, 205, 208, 211, 214, 217, 220, 223, 226,
229, 232, 235, 238,
  241, 244, 247, 250, 253, 256, 259, 262, 265, 268, 271, 274, 277, 280,
283, 286, 289, 292,
  295, 298, 301, 304}

{2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53,
56, 59, 62, 65, 68, 71,
  74, 77, 80, 83, 86, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119,
122, 125, 128, 131,
  134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173,
176, 179, 182, 185,
  188, 191, 194, 197, 200, 203, 206, 209, 212, 215, 218, 221, 224, 227,
230, 233, 236, 239,
  242, 245, 248, 251, 254, 257, 260, 263, 266, 269, 272, 275, 278, 281,
284, 287, 290, 293,
  296, 299, 302, 305}

{}

and after 10^4 iterations (without intersections), lists end :

In[121]:=
Take[#1, -5]& /@ {list1, list2}

Out[121]=
{{29992, 29995, 29998, 30001, 30004},
 {29993, 29996, 29999, 30002, 30005}}

This time we've got {x: x == i mod 3} with i = 1,2.

Testing the 10^4 iterations:

Union[Mod[#,3]]&/@{list1,list2}
--> {{1},{2}}

Could you please explain in detail, how you've got your sequences?

Peter




From maxale at gmail.com  Tue Jul 31 05:47:31 2007
From: maxale at gmail.com (Max Alekseyev)
Date: Mon, 30 Jul 2007 20:47:31 -0700
Subject: Do any integers occur in both sequences?
In-Reply-To: <46AEA87E.1010709@dordos.net>
References: <E1IFWBz-0000Pz-00@pop03.mail.atl.earthlink.net>
	 <46AEA87E.1010709@dordos.net>
Message-ID: <d3dac270707302047i346165cred3b11e98db90bcd@mail.gmail.com>

On 7/30/07, Peter Pein <petsie at dordos.net> wrote:
> Leroy Quet schrieb:
> > I have just submitted these two interdependent sequences (So don't look
> > for them in the database yet):
> >
> >> %I A131937
> >> %S A131937 1,4,8,14,21,29,38,49,61
> >> %N A131937 a(1)=1; a(2)=4. a(n) = a(n-1) + (nth positive integer which
> >> does not occur in sequence A131938).
> >> %e A131937 A131938: 2,5,10,16,23,32,42,53,...
> >> Positive integers not in A131938: 1,3,4,6,7,8,9,11,...
> >> So A131937(8) = A131937(7) + 11 = 49.
> >> %Y A131937 A131938
> >> %O A131937 1
> >> %K A131937 ,more,nonn,
> >
> >> %I A131938
> >> %S A131938 2,5,10,16,23,32,42,53,65,78,93,109
> >> %N A131938 a(1)=2; a(2)=5. a(n) = a(n-1) + (nth positive integer which
> >> does not occur in sequence A131937).
> >> %e A131938 A131937: 1,4,8,14,21,29,...
> >> Positive integers not in A131937: 2,3,5,6,7,9,10,11,...
> >> So A131938(8) = A131938(7) + 11 = 53.
> >> %Y A131938 A131937
> >> %O A131938 1
> >> %K A131938 ,more,nonn,

[...]

> Could you please explain in detail, how you've got your sequences?

This is my PARI/GP code (for the first 50 terms) that confirms Leroy's values:

{
A=Set([1,4]); B=Set([2,5]);
a=4; na=4;
b=5; nb=3;
for(n=3,50,
  until(!setsearch(A,na),na++);
  until(!setsearch(B,nb),nb++);
  a+=nb;
  b+=na;
  A=setunion(A,[a]);
  B=setunion(B,[b]);
);
print(vecsort(eval(A)));
print(vecsort(eval(B)));
}

Here the sets A and B collect elements (as computed) of A131937 and
A131938 respectively; the variables a and b go over elements of A and
B; variables na and nb go over non-elements of A and B.

The output is:

[1, 4, 8, 14, 21, 29, 38, 49, 61, 74, 88, 103, 120, 138, 157, 177,
198, 220, 244, 269, 295, 322, 350, 379, 409, 440, 473, 507, 542, 578,
615, 653, 692, 732, 773, 816, 860, 905, 951, 998, 1046, 1095, 1145,
1196, 1248, 1302, 1357, 1413, 1470, 1528]

[2, 5, 10, 16, 23, 32, 42, 53, 65, 78, 93, 109, 126, 144, 163, 183,
205, 228, 252, 277, 303, 330, 358, 388, 419, 451, 484, 518, 553, 589,
626, 665, 705, 746, 788, 831, 875, 920, 966, 1013, 1061, 1111, 1162,
1214, 1267, 1321, 1376, 1432, 1489, 1547]

Regards,
Max



A018892 ("Number of ways to write 1/n as a sum of exactly 2 unit fractions")
has a simple formula a(n) = (d(n^2) + 1)/2, as noted in the first comment.

There is a simple construction not mentioned there that nicely demonstrates
the formula: 1/n = 1/(n+a) + 1/(n+b) implies ab = n^2.

The same construction extends to the general case: to write m/n as a sum
of 2 unit fractions, find a factorisation n^2 = xy, such that
n + x == n + y == 0 (mod m). Then m/n = m/(n+x) + m/(n+y), and the RHS

When m = 2, it remains simple: for odd n, any pair of factors of n^2 will
both be odd, so n+x, n+y will be even in every case. So given:

b(n) = Number of ways to write 2/n as a sum of exactly 2 unit fractions
we get:
b(n) = { A018892(n/2)  if n == 0 (mod 2)

For c(n) = Number of ways to write 3/n as a sum of exactly 2 unit fractions,
we need to find the number of ways to split n^2 into 2 factors each
equivalent either to 1 (mod 3) when n == 2 (mod 3), or vice versa.

Writing n = PQ, such that P is a product of primes == 1 (mod 3) and
Q a product of primes == 2 (mod 3), I find:

c(n) = { A018892(n/3)               if n == 0 (mod 3)

I have an approach to prove this, by induction on multiplication of prime
powers, but I suspect it is overly complex: can anyone suggest a simple
proof of c(n)?

I think the 2/n and 3/n cases are interesting enough to submit once I've
for other m, I don't expect to submit those unless they prove unexpectedly
interesting.

Hugo




From hv at crypt.org  Tue Jul 31 14:01:16 2007
From: hv at crypt.org (hv at crypt.org)
Date: Tue, 31 Jul 2007 13:01:16 +0100
Subject: A018892 and extensions
Message-ID: <200707311201.l6VC1GwJ004230@zen.crypt.org>

simplifies to two unit fractions.
       { A018892(n)    if n == 1 (mod 2)
       { d(P^2) (d(Q^2) - 1) / 4    if n == 1 (mod 3)
       { d(P^2) (d(Q^2) + 1) / 4    if n == 2 (mod 3)
satisfied myself of their correctness; while I plan to investigate m/n
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Max Alekseyev schrieb:
> On 7/28/07, Peter Pein <petsie at dordos.net> wrote:
> 
>>> Each here line contains a pair:
>>> n k
>>> such that sigma(3,k)=sigma(3,n) and k=a(n) for r=3 (following your
>>> notations above).
>>>
>>> I will let my program to run for a couple more days to reach 10^8 bound.
> 
> I stopped the program after it reached 7*10^7 as it started to eat too
> much memory.
> The results are attached. Please format them and submit to OEIS as you
> feel appropriate.
> 
> P.S. Eventually I've defended Ph.D. in Computer Science ;)

... if you felt that this has been necessary... ;-)
> 
> Regards,
> Max
> 
> 
Thank you again, Max.

As soon as I've got a little bit more time to spare, I'll enter the values

Peter




From petsie at dordos.net  Tue Jul 31 15:18:16 2007
From: petsie at dordos.net (Peter Pein)
Date: Tue, 31 Jul 2007 15:18:16 +0200
Subject: Do any integers occur in both sequences?
In-Reply-To: <46AEA87E.1010709@dordos.net>
References: <E1IFWBz-0000Pz-00@pop03.mail.atl.earthlink.net> <46AEA87E.1010709@dordos.net>
Message-ID: <46AF3698.6080307@dordos.net>

Well, I should not post as early in the morning (at least not before
I've got the third cup of coffee). I did not add a(n-1)....

Sorry for any inconvenience, trouble and the like

Peter

P.S.:
A131938-A131937 is (seems to be after coffee ;) ):

{1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6,
  7, 8, 8, 8, 8, 8, 8, 9, 10, 11, 11, 11, 11,
  11, 11, 12, 13, 14, 15, 15, 15, 15, 15, 15,
  15, 16, 17, 18, 19, 19, 19, 19, 19, 19, 19,
  19, 20, 21, 22, 23, 23, 23, 23, 23, 23, 23,
  23, 23, 24, 25, 26, 27, 27, 27, 27, 27, 27,
  27, 27, 27, 27, 28, 29, 30, 31, 32, 32, 32,
  32, 32, 32, 32, 32, 32, 32, 33, 34, 35, 36,
  37, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38,
  38, 39, 40, 41, 42, 43, 44, 44, 44, 44, 44,
  44, 44, 44, 44, 44, 44, 44, 45, 46, 47, 48,
  49, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50,
  50, 50, 50, 51, 52, 53, 54, 55, 56, 56, 56,
  56, 56}

Peter Pein schrieb:
> Leroy Quet schrieb:
>> I have just submitted these two interdependent sequences (So don't look 
>> for them in the database yet):
>>
>>> %I A131937
>>> %S A131937 1,4,8,14,21,29,38,49,61
>>> %N A131937 a(1)=1; a(2)=4. a(n) = a(n-1) + (nth positive integer which 
>>> does not occur in sequence A131938).
>>> %e A131937 A131938: 2,5,10,16,23,32,42,53,...
>>> Positive integers not in A131938: 1,3,4,6,7,8,9,11,...
>>> So A131937(8) = A131937(7) + 11 = 49.
>>> %Y A131937 A131938
>>> %O A131937 1
>>> %K A131937 ,more,nonn,
>>> %I A131938
>>> %S A131938 2,5,10,16,23,32,42,53,65,78,93,109
>>> %N A131938 a(1)=2; a(2)=5. a(n) = a(n-1) + (nth positive integer which 
>>> does not occur in sequence A131937).
>>> %e A131938 A131937: 1,4,8,14,21,29,...
>>> Positive integers not in A131937: 2,3,5,6,7,9,10,11,...
>>> So A131938(8) = A131938(7) + 11 = 53.
>>> %Y A131938 A131937
>>> %O A131938 1
>>> %K A131938 ,more,nonn,
>>
>> I have not thought about this too hard; so for all I know, the proof is 
>> quite easy.
>>
>> Do any positive integers occur in both A131937 and A131938?
>>
>> Thanks,
>> Leroy Quet
>>
> Hi again,
> 
> even if you meant "n-th pos. int. which does yet not occur in either l1
> or l2" (or does one say "..does yet neither occur in l1 nor in l2"?) I
> get another result
> 
> list1={1,4};list2={2,5};n=3;
> Do[
>     Print[{list1,list2}];
>     Print["n= ",n," ",Complement[Range[25],list2]];
>     AppendTo[list1,Part[Complement[Range[3n ],Union[list1,list2]],n]];
>     Print["n= ",n," ",Complement[Range[25],list1]];
>     AppendTo[list2,Part[Complement[Range[3n],Union[list1,list2]],n]];
>     n++;
>     ,{5}];
> list1
> list2
> Intersection[%%,%]
> 
....



I wrote:
>I have just submitted these two interdependent sequences (So don't look 
>for them in the database yet):
>
>>%I A131937
>>%S A131937 1,4,8,14,21,29,38,49,61
>>%N A131937 a(1)=1; a(2)=4. a(n) = a(n-1) + (nth positive integer which 
>>does not occur in sequence A131938).
>>%e A131937 A131938: 2,5,10,16,23,32,42,53,...
>>Positive integers not in A131938: 1,3,4,6,7,8,9,11,...
>>So A131937(8) = A131937(7) + 11 = 49.
>>%Y A131937 A131938
>>%O A131937 1
>>%K A131937 ,more,nonn,
>
>>%I A131938
>>%S A131938 2,5,10,16,23,32,42,53,65,78,93,109
>>%N A131938 a(1)=2; a(2)=5. a(n) = a(n-1) + (nth positive integer which 
>>does not occur in sequence A131937).
>>%e A131938 A131937: 1,4,8,14,21,29,...
>>Positive integers not in A131937: 2,3,5,6,7,9,10,11,...
>>So A131938(8) = A131938(7) + 11 = 53.
>>%Y A131938 A131937
>>%O A131938 1
>>%K A131938 ,more,nonn,
>
>I have not thought about this too hard; so for all I know, the proof is 
>quite easy.
>
>Do any positive integers occur in both A131937 and A131938?
>
>Thanks,
>Leroy Quet

I have made no progress towards determining if any particular positive 
integers occur within both sequences. (In other words: Does  A131937(k) = 
A131938(j) for any j and k {j and k are >= 1}, where j need not equal k?)

I conjecture that no particular positive integer occurs in both sequences.


Here is a smaller result related to these sequences, which I doubt will 
help (dis)prove the main conjecture:

Let a(n) =  A131937(n), b(n) = A131938(n), a(0) = b(0) = 0.

Let n = any positive integer.

Then n occurs (a(n) - a(n-1) - 1) times in sequence {b(n) - b(n-1) - n + 
1}.

And n occurs (b(n) - b(n-1) - 1) times in sequence {a(n) - a(n-1) - n + 
1}.

Not too earth-shattering -- but while we're on the subject...


Also, I wonder if anyone can come up with a closed form for {A131937(n)} 
and {A131938(n)}.
They seem like they might be related to Beautty sequences somehow.

Thanks,
Leroy Quet



A comment with seq A001006
http://www.research.att.com/~njas/sequences/A001006

This seems to be wrong, I get the counts

Note the starts math, Motzkins start as


Someone please verify!





From qq-quet at mindspring.com  Tue Jul 31 16:18:48 2007
From: qq-quet at mindspring.com (Leroy Quet)
Date: Tue, 31 Jul 07 08:18:48 -0600
Subject: Do any integers occur in both sequences?
Message-ID: <E1IFsbM-0006LJ-00@pop05.mail.atl.earthlink.net>

says (near top):
  Number of sequences of length n-1 consisting of positive integers
  such that the opening and ending elements are 1 or 2, and the
  absolute difference between any 2 consecutive elements is 0 or 1.
 A024537 ,1, 2, 4, 9, 21, 50, 120, 289, 697, 1682 (=?= A018905)
     [1,] 1, 2, 4, 9, 21, 51, 127, 323,
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Date: Tue, 31 Jul 2007 10:50:49 -0700
From: "Max Alekseyev" <maxale at gmail.com>
To: "Joerg Arndt" <arndt at jjj.de>
Subject: Re: possibly wrong comment with Motzkin numbers
Cc: seqfan at ext.jussieu.fr
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On 7/31/07, Joerg Arndt <arndt at jjj.de> wrote:
> A comment with seq A001006
> http://www.research.att.com/~njas/sequences/A001006
> says (near top):
>   Number of sequences of length n-1 consisting of positive integers
>   such that the opening and ending elements are 1 or 2, and the
>   absolute difference between any 2 consecutive elements is 0 or 1.
>
> This seems to be wrong, I get the counts
>  A024537 ,1, 2, 4, 9, 21, 50, 120, 289, 697, 1682 (=?= A018905)
>
> Note the starts math, Motzkins start as
>      [1,] 1, 2, 4, 9, 21, 51, 127, 323,

With a simple recurrence formula implementation in PARI/GP, I confirm
that the comment in A001006 is correct:

{ a(n,l) = if(n==1,(l==1)||(l==2),if(l<=0,0,a(n-1,l-1)+a(n-1,l)+a(n-1,l+1))) }

{ f(n) = a(n,1)+a(n,2) }

? vector(10,n,f(n))
%1 = [2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798]

Here a(n,l) counts the number of sequences of positive integers of
length n, starting with 1 or 2 and ending with l.

Max