[seqfan] R: Seqfan Digest, Vol 10, Issue 5
vincenzo.librandi at tin.it
vincenzo.librandi at tin.it
Mon Jul 6 21:40:43 CEST 2009
----Messaggio originale----
Da: seqfan-request at list.seqfan.eu
Data: 6-lug-2009 2.00 PM
A: <seqfan at list.seqfan.eu>
Ogg: Seqfan Digest, Vol 10, Issue 5
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Today's Topics:
1. Re: Plot Of A160855 (franktaw at netscape.net)
2. Re: Another BBP formula for ln(2) ? (Simon Plouffe)
3. Re: Another BBP formula for ln(2) ? (Simon Plouffe)
4. Re: Destinies of sums of proper divisors. (Hans Havermann)
5. Re: Destinies of sums of proper divisors. (franktaw at netscape.net)
6. Re: Destinies of sums of proper divisors. (Jon Awbrey)
7. Re: Destinies of sums of proper divisors. (Hans Havermann)
8. sequence (vincenzo.librandi at tin.it)
9. A134063 and A162399 are the same (Prof. Dr. Alois Heinz)
10. Re: Destinies of sums of proper divisors. (David Wilson)
11. Re: Destinies of sums of proper divisors. (David Wilson)
----------------------------------------------------------------------
Message: 1
Date: Sun, 05 Jul 2009 16:13:35 -0400
From: franktaw at netscape.net
Subject: [seqfan] Re: Plot Of A160855
To: seqfan at list.seqfan.eu, seqfan at seqfan.eu
Message-ID: <8CBCBC435FC0E42-EE0-295A at webmail-me07.sysops.aol.com>
Content-Type: text/plain; charset="us-ascii"; format=flowed
The first step in answering this kind of question is to determine what
those lines are. What are the slopes and intercepts? If you have
that, it might be easy to see why they appear. This is something that
you can and should do for yourself, Leroy.
Franklin T. Adams-Watters
-----Original Message-----
From: Leroy Quet <q1qq2qqq3qqqq at yahoo.com>
A160855 is:
a(n) = the smallest positive integer not occurring earlier in the
sequence such
that sum{k=1 to n} a(k) written in binary contains binary n as a
substring.
(1, 3, 2, 6, 8, 4, 5, 11, 10, 24, 12, 13, 7, 9, 28, 17, 36, 14, 20,
46,...)
Now, after Ray Chandler, I think it was, extended this sequence out for
many
terms, its scatter-plot was at that point rather interesting, in my
opinion.
But now that H. v. Eitzen has extended the sequence out to 100000
terms, the
scatter-plot is even more interesting.
What explains the radiating lines, or explains the seeming random blobs
seen
when the scatter-plot is taken to a smaller number of terms?
I must be missing something obvious as to why these structures occur.
What kinds
of structures are seen when the plot is taken even farther out, I
wonder?
Thanks,
Leroy Quet
------------------------------
Message: 2
Date: Mon, 06 Jul 2009 00:55:51 +0200
From: Simon Plouffe <simon.plouffe at gmail.com>
Subject: [seqfan] Re: Another BBP formula for ln(2) ?
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Cc: seqfan at seqfan.eu
Message-ID: <4A512F77.4080803 at gmail.com>
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Hello,
yes, it is true, by changing some variable, there is
several ways to get a more efficient formula than
log(2) = sum(1/n/2^n,n=1..infinity), (formula 1)
a question : what is C in equation 0.135 ? it must be Pi
isn't ?
For Alexander, look at equation 0.129 and 0.130 of
mr Mathar paper, these formulas are with the index 16^n :
it means that , formulas with an index of 2^n and 10 or
more terms are of very little interest since it converges
not faster than the standard log(2) series of Euler.
If I remember well, the formula (1) can be obtained
by using EULER transform of the series 1 - 1/2 + 1/3 - 1/4 ...= log(2).
Since there are many series of that later type : many others
can be obtained.
Also, and finally, there is always a way to SPLIT into 2 parts
a series of harmonic type to obtain a formula S = S1 + S2 and
deduce a NEW formula more rapidly convergent and for log(2)
there is a plethora of ways to get a better index than of 2^n.
best regards,
Simon Plouffe
------------------------------
Message: 3
Date: Mon, 06 Jul 2009 00:55:51 +0200
From: Simon Plouffe <simon.plouffe at gmail.com>
Subject: [seqfan] Re: Another BBP formula for ln(2) ?
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Cc: seqfan at seqfan.eu
Message-ID: <4A512F77.4080803 at gmail.com>
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Hello,
yes, it is true, by changing some variable, there is
several ways to get a more efficient formula than
log(2) = sum(1/n/2^n,n=1..infinity), (formula 1)
a question : what is C in equation 0.135 ? it must be Pi
isn't ?
For Alexander, look at equation 0.129 and 0.130 of
mr Mathar paper, these formulas are with the index 16^n :
it means that , formulas with an index of 2^n and 10 or
more terms are of very little interest since it converges
not faster than the standard log(2) series of Euler.
If I remember well, the formula (1) can be obtained
by using EULER transform of the series 1 - 1/2 + 1/3 - 1/4 ...= log(2).
Since there are many series of that later type : many others
can be obtained.
Also, and finally, there is always a way to SPLIT into 2 parts
a series of harmonic type to obtain a formula S = S1 + S2 and
deduce a NEW formula more rapidly convergent and for log(2)
there is a plethora of ways to get a better index than of 2^n.
best regards,
Simon Plouffe
------------------------------
Message: 4
Date: Sun, 5 Jul 2009 20:46:48 -0400
From: Hans Havermann <pxp at rogers.com>
Subject: [seqfan] Re: Destinies of sums of proper divisors.
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Message-ID: <541BA737-E196-4D5B-A82D-63E591B780E7 at rogers.com>
Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes
Maximilian Hasler:
> I must admit that dest(99225) still takes some time to compute...
Indeed. The destiny is identical to that of the smaller 46758:
0 46758 99225
1 46770 114582
2 65550 135762
3 113010 180462
4 158286 199698
5 191922 205518
6 205518 ...
so I looked for dest(46758), first using Mathematica to #472, then one
at a time (copy/paste) using Dario Alpern's online ECM-factorization
applet to #547 (< #546). #547 (assuming I didn't make any mistakes) is:
220856107213904390193665654934000814055972896201951406931845871108450633171297110864659603360
According to Wolfgang Creyaufmueller's online tables
http://www.aliquot.de/tabellen/tables.htm
46758 has been indexed up to #666 where we have the 101-digit
42291149432355095308926992983480450623448873555059382537140396682361170391404079245409226419855814760
------------------------------
Message: 5
Date: Sun, 05 Jul 2009 21:33:20 -0400
From: franktaw at netscape.net
Subject: [seqfan] Re: Destinies of sums of proper divisors.
To: seqfan at list.seqfan.eu
Message-ID: <8CBCBF0E12D5F0E-1450-23E at webmail-me07.sysops.aol.com>
Content-Type: text/plain; charset="us-ascii"; format=flowed
Please don't call this the "destiny". Destiny implies where you wind
up. "First decrease" would be fine, if nobody comes up with something
better.
Franklin T. Adams-Watters
-----Original Message-----
From: Maximilian Hasler <maximilian.hasler at gmail.com>
I suggest a different notion of "destiny"
(or someone may find a better name),
namely, the first term in the trajectory under
s(n)=sigma(n)-n
which is not larger than its predecessor:
...
------------------------------
Message: 6
Date: Sun, 05 Jul 2009 21:42:08 -0400
From: Jon Awbrey <jawbrey at att.net>
Subject: [seqfan] Re: Destinies of sums of proper divisors.
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Message-ID: <4A515670.8080701 at att.net>
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
"Los Lieder"
franktaw at netscape.net wrote:
> Please don't call this the "destiny". Destiny implies where you wind up.
> "First decrease" would be fine, if nobody comes up with something better.
>
> Franklin T. Adams-Watters
--
inquiry list: http://stderr.org/pipermail/inquiry/
mwb: http://www.mywikibiz.com/Directory:Jon_Awbrey
knol: http://knol.google.com/k/-/-/3fkwvf69kridz/1
------------------------------
Message: 7
Date: Sun, 5 Jul 2009 22:59:52 -0400
From: Hans Havermann <pxp at rogers.com>
Subject: [seqfan] Re: Destinies of sums of proper divisors.
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Message-ID: <9C080EFF-99B5-47FE-94D0-38DFBD4FDFD7 at rogers.com>
Content-Type: text/plain; charset=US-ASCII; format=flowed
>> the first term in the trajectory under
>> s(n)=sigma(n)-n
>> which is not larger than its predecessor
> "First decrease" would be fine...
"First non-increase" would be more accurate.
------------------------------
Message: 8
Date: Mon, 6 Jul 2009 14:24:29 +0200 (CEST)
From: "vincenzo.librandi at tin.it" <vincenzo.librandi at tin.it>
Subject: [seqfan] sequence
To: seqfan at list.seqfan.eu
Cc: vincenzo.librandi at tin.it
Message-ID:
<28410447.361791246883069253.JavaMail.defaultUser at defaultHost>
Content-Type: text/plain; charset=UTF-8
Hi all
Ask you formula sequence:
3, 3, 3, 4, 4, 5, 5, 6, 8, 8, 10, 11, 11, 12, 14, 16, 16, 18, 19, 19, 21, 22, 24, 27, 28, 28, 29, 29, 30, 36, 37 ,...,
or formula sequence:
0, 0, 0, 1, 1, 2, 2, 3, 5, 5, 7, 8, 8, 9, 11, 13, 13, 15, 16, 16, 18, 19, 21, 24, 25, 25, 26, 26, 27, 33,...,
quando ai termini della prima sequenza si sottrae 3.
Best regards,
Vincenzo Librandi
===================
Thans fot Hasler and Mathar.
OK A008507, I now it.
volevo però una sequenza senza utilizzare i numeri prmi.
Lo scopo:
If K = A067076 (0,1,2,4,5,7,8,10, and so on) tali che 2*k+3=primo
If S_K =A008507 (3,3,3,4,4,5,5,6,8, an so on)
che chiamo "Salti di K", infatti per
K S_k
0 3
1 3 (nessun salto da 0 a 1)
2 3
4 4 (un salto perchè manca il 3)
5 4
7 5 (un salto, manca 6)
8 5
10 6
13 8 (due salti perch mancano 11e 12)
14 8
17 10 (due salti perchè 15 e 16)
19 11
and so on
se alle due colonne aggiungiamo la serie dei numeri naturali (0,1,2,3,4,5,6,7,, ..) avremo
0+3+0=3
1+3+1=5
2+3+2=7
4+4+3=11
5+4+4=13
7+5+5=17
8+5+6=19
10+6+7=23
and so on
Come potremo legare il tutto ?
Thanks,
Vincenzo Librandi
------------------------------
Message: 9
Date: Mon, 06 Jul 2009 15:09:53 +0200
From: "Prof. Dr. Alois Heinz" <heinz at hs-heilbronn.de>
Subject: [seqfan] A134063 and A162399 are the same
To: seqfan at seqfan.eu
Cc: rlahaye at new.rr.com, "N. J. A. Sloane" <njas at research.att.com>,
the at iwu.edu
Message-ID: <4A51F7A1.7000606 at hs-heilbronn.de>
Content-Type: text/plain; charset=ISO-8859-15; format=flowed
Both sequences are essentially the same:
http://www.research.att.com/~njas/sequences/A134063
http://www.research.att.com/~njas/sequences/A162399
I propose to delete the latter.
Alois
------------------------------
Message: 10
Date: Mon, 06 Jul 2009 09:25:28 -0400
From: David Wilson <dwilson at gambitcomm.com>
Subject: [seqfan] Re: Destinies of sums of proper divisors.
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Message-ID: <4A51FB48.5020807 at gambitcomm.com>
Content-Type: text/plain; charset=us-ascii; format=flowed
I believe the proper terms are:
"trajectory of n" for the sequence
t(k) =
n if k = 0
f(t(k-1)) if k > 0
"attractor" for a set S with f(S) = S. (that is, the set of values in a
trajectory loop).
"fixpoint" for a single-element attractor (that is, {n} or n with f(n) = n).
"stopping point" for an element satisfying an artificial "stopping
condition", such as n = 1 for Collatz or f(n+1) <= n for the sequence in
question.
So "stopping point with condition f(n+1) <= n" would avoid having to
coin new terms.
franktaw at netscape.net wrote:
> Please don't call this the "destiny". Destiny implies where you wind
> up. "First decrease" would be fine, if nobody comes up with something
> better.
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: Maximilian Hasler <maximilian.hasler at gmail.com>
>
> I suggest a different notion of "destiny"
> (or someone may find a better name),
> namely, the first term in the trajectory under
> s(n)=sigma(n)-n
> which is not larger than its predecessor:
>
> ...
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
------------------------------
Message: 11
Date: Mon, 06 Jul 2009 09:58:43 -0400
From: David Wilson <dwilson at gambitcomm.com>
Subject: [seqfan] Re: Destinies of sums of proper divisors.
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Message-ID: <4A520313.30800 at gambitcomm.com>
Content-Type: text/plain; charset=us-ascii; format=flowed
I also wanted to make this observation on aliquot sequences:
Let s(n) be the sum of divisors of n, and f(n) = s(n)-n be the sum of
aliquot divisors. We are discussing the trajectory of f(n).
Note that if n is a large number of the form 28k with k odd, then f(n)
is likely to be a yet larger number of this form. If a sufficiently
large number of this form occurs in the trajectory of n, the subsequent
trajectory is likely retain that form and grow. Once it has grown
sufficiently large, in the unlikely event it loses the form 28k, it is
likely to recover that form after a few iterations a few iterations of
and resume its growth.
For this reason, I conjecture that f(n) is very likely to include
divergent trajectories, and that n = 276 is probably one of them.
------------------------------
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