From njas at research.att.com  Tue Dec 11 18:29:14 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Tue, 11 Dec 2001 12:29:14 -0500 (EST)
Subject: sum*product = n
Message-ID: <200112111729.MAA73530@fry.research.att.com>

Sol Golomb recently mentioned an old puzzle:

find all n such that sum of digits of n times
product of digits of n  is equal to n

this is a finite sequence which begins 1, 135, 144

does any seq fan know of further terms?

Neil

e g   (1+3+5)*1*3*5) = 135




From edp at wolfram.com  Tue Dec 11 18:57:20 2001
From: edp at wolfram.com (Ed Pegg Jr)
Date: Tue, 11 Dec 2001 11:57:20 -0600
Subject: Wilson's Corollary
In-Reply-To: <200112111729.MAA73530@fry.research.att.com>
Message-ID: <DCEIJGPGGGDPBLNBEBNBEEAACNAA.edp@wolfram.com>


  Last night, I looked at Wilson's Corollary.  I wrote
up the results at www.mathpuzzle.com .  There are many
sequences in this.  So far, I haven't found any in EIS,
but perhaps I'm looking in the wrong area.

  --Ed Pegg Jr.




From njas at research.att.com  Tue Dec 11 19:07:36 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Tue, 11 Dec 2001 13:07:36 -0500 (EST)
Subject: sum*product POSTSCRIPT
Message-ID: <200112111807.NAA26480@fry.research.att.com>

i should have looked harder 

this is A038369, which is labeled as "fini,full"

please ignore previous posting!




From rkg at cpsc.ucalgary.ca  Tue Dec 11 19:48:58 2001
From: rkg at cpsc.ucalgary.ca (Richard Guy)
Date: Tue, 11 Dec 2001 11:48:58 -0700 (MST)
Subject: sum*product POSTSCRIPT
In-Reply-To: <200112111807.NAA26480@fry.research.att.com>
Message-ID: <Pine.LNX.4.33.0112111138150.13879-100000@csl>

A small glitch in the following, which shd have 10^k - 1
instead of 10^(k-1).  This enables one to see that k < 60.
Still a bit big for a brute force search!
Note that if 5 occurs, then all digits are odd.
Also that, except possibly for one factor of size
< 9k, the number  n  is 7-smooth.       R.

%I A038369
%S A038369 0,1,135,144
%N A038369 n = (product of digits of n) * (sum of digits of n).
%C A038369 To prove finiteness, let k be the number of digits and consider 9^k*9*k < 10^(k-1) for every k > 100 - Ulrich Schimke (UlrSchimke at aol.com).
%H A038369 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Sum-ProductNumber.html">Link to a section of The World of Mathematics.</a>
%H A038369 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Digit.html">Link to a section of The World of Mathematics.</a>
%e A038369 144 belongs to the sequence because 1*4*4=16, 1+4+4=9 -> 16*9=144
%Y A038369 n = A007953(n) * A007954(n).
%K A038369 nice,nonn,fini,base,full,bref
%O A038369 1,3
%A A038369 Felice Russo (felice.russo at katamail.com)

On Tue, 11 Dec 2001, N. J. A. Sloane wrote:

> i should have looked harder 
> 
> this is A038369, which is labeled as "fini,full"





From njas at research.att.com  Tue Dec 11 20:10:20 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Tue, 11 Dec 2001 14:10:20 -0500 (EST)
Subject: sum*product POSTSCRIPT
Message-ID: <200112111910.OAA47069@fry.research.att.com>

Richard, I think 10^(k-1) was correct - the point being
that this is a LOWER bound on n.

anyway, i've replaced that comment with David Wilson's proof
that the sequence is complete





From rgwv at kspaint.com  Tue Dec 11 20:16:49 2001
From: rgwv at kspaint.com (Robert G. Wilson v)
Date: Tue, 11 Dec 2001 13:16:49 -0600
Subject: sum*product = n
References: <200112111729.MAA73530@fry.research.att.com>
Message-ID: <3C165BA1.EA55B4DA@kspaint.com>

Et al,

        Using Mathematica, I have tested the sequence up to 7.5*10^7 and
have found no further numbers which satisfy the criteria.

Bob.

"N. J. A. Sloane" wrote:

> Sol Golomb recently mentioned an old puzzle:
>
> find all n such that sum of digits of n times
> product of digits of n  is equal to n
>
> this is a finite sequence which begins 1, 135, 144
>
> does any seq fan know of further terms?
>
> Neil
>
> e g   (1+3+5)*1*3*5) = 135





From wilson at aprisma.com  Tue Dec 11 21:58:32 2001
From: wilson at aprisma.com (David W. Wilson)
Date: Tue, 11 Dec 2001 15:58:32 -0500
Subject: sum*product = n
References: <200112111729.MAA73530@fry.research.att.com>
    <3C165BA1.EA55B4DA@kspaint.com>
Message-ID: <3C167378.B424581@aprisma.com>

"Robert G. Wilson v" wrote:
> 
> Et al,
> 
>         Using Mathematica, I have tested the sequence up to 7.5*10^7 and
> have found no further numbers which satisfy the criteria.
> 
> Bob.
> 
> "N. J. A. Sloane" wrote:
> 
> > Sol Golomb recently mentioned an old puzzle:
> >
> > find all n such that sum of digits of n times
> > product of digits of n  is equal to n
> >
> > this is a finite sequence which begins 1, 135, 144
> >
> > does any seq fan know of further terms?
> >
> > Neil
> >
> > e g   (1+3+5)*1*3*5) = 135

I showed that 1, 135, and 144.  See "Sum-Product Number" at mathworld
for details.




From wilson at aprisma.com  Tue Dec 11 22:00:24 2001
From: wilson at aprisma.com (David W. Wilson)
Date: Tue, 11 Dec 2001 16:00:24 -0500
Subject: sum*product = n
References: <200112111729.MAA73530@fry.research.att.com>
    <3C165BA1.EA55B4DA@kspaint.com>
Message-ID: <3C1673E8.B1F54693@aprisma.com>

"Robert G. Wilson v" wrote:
> 
> Et al,
> 
>         Using Mathematica, I have tested the sequence up to 7.5*10^7 and
> have found no further numbers which satisfy the criteria.
> 
> Bob.
> 
> "N. J. A. Sloane" wrote:
> 
> > Sol Golomb recently mentioned an old puzzle:
> >
> > find all n such that sum of digits of n times
> > product of digits of n  is equal to n
> >
> > this is a finite sequence which begins 1, 135, 144
> >
> > does any seq fan know of further terms?
> >
> > Neil
> >
> > e g   (1+3+5)*1*3*5) = 135

Try again.

I showed that 1, 135, and 144 are all there are. 

See "Sum-Product Numbers" at mathworld.com.

- Dave Wilson
  Alzheimer's victim




From m.v.subbarao at ualberta.ca  Wed Dec 12 20:31:47 2001
From: m.v.subbarao at ualberta.ca (M.V Subbarao)
Date: Wed, 12 Dec 2001 12:31:47 -0700 (MST)
Subject: sumtimes product dividing n
Message-ID: <Pine.A41.4.33.0112121217190.31460-100000@gpu1.srv.ualberta.ca>


I wonder if there are infinitely many integers n such that the sum of its
digits times product of its digits is a divisor of n. On  example is n =
12. No doubt, we  can easily find many more.

M.V.Subbarao
m.v.subbarao at ualberta.ca





From rgwv at kspaint.com  Wed Dec 12 20:42:46 2001
From: rgwv at kspaint.com (Robert G. Wilson v)
Date: Wed, 12 Dec 2001 13:42:46 -0600
Subject: sumtimes product dividing n
References: <Pine.A41.4.33.0112121217190.31460-100000@gpu1.srv.ualberta.ca>
Message-ID: <3C17B335.52456749@kspaint.com>

Dear M.V.

        See
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=049102

Sincerely yours,
Robert G. Wilson, V

"M.V Subbarao" wrote:

> I wonder if there are infinitely many integers n such that the sum of its
> digits times product of its digits is a divisor of n. On  example is n =
> 12. No doubt, we  can easily find many more.
>
> M.V.Subbarao
> m.v.subbarao at ualberta.ca





From njas at research.att.com  Wed Dec 12 20:50:51 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Wed, 12 Dec 2001 14:50:51 -0500 (EST)
Subject: Jeroboam
Message-ID: <200112121950.OAA33220@fry.research.att.com>

A couple of weeks ago Marc LeBrun said

>  >=N. J. A. Sloane
>  > ...sequence number A065536...
> 
> Excellent!
> 
> In honor of this event, and in "thanksgiving" for your sterling
> proprietorship, on behalf of everyone I've taken the liberty of sending you
> a small token of all of our appreciation.  (Reassure your mail room; we're
> merely fanatics, not "sequence terrorists"!).
> 

I had forgotten all about this message but today the mail room
delivered one jeroboam of 21st Century Mumm Cuvee Napa.
Pretty amazing - never seen a jeroboam before
that wasn't locked up in a glass case.

Too bad the sequence fanatics are scattered over the 
whole world, otherwise we could have a party.  Email can't
do everything!

Thanks, Marc - and best wishes to everyone for 2002

Neil




From jawbrey at oakland.edu  Wed Dec 12 21:08:32 2001
From: jawbrey at oakland.edu (Jon Awbrey)
Date: Wed, 12 Dec 2001 15:08:32 -0500
Subject: Jeroboam
References: <200112121950.OAA33220@fry.research.att.com>
Message-ID: <3C17B940.88587978@oakland.edu>

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

|         ?
|        ?
|         .?
|        \_/
|         |
|        -^-
|
| e-bullient e-bubbly
|
| and cheers to all ...
|
| jon awbrey

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

N. J. A. Sloane wrote:
> 
> A couple of weeks ago Marc LeBrun said
> 
> > > N.J.A. Sloane
> > > ...sequence number A065536...
> >
> > Excellent!
> >
> > In honor of this event, and in "thanksgiving" for your sterling proprietorship,
> > on behalf of everyone I've taken the liberty of sending you a small token of
> > all of our appreciation.  (Reassure your mail room;  we're merely fanatics,
> > not "sequence terrorists"!).
> 
> I had forgotten all about this message but today the mail room
> delivered one jeroboam of 21st Century Mumm Cuvee Napa.
> Pretty amazing - never seen a jeroboam before
> that wasn't locked up in a glass case.
> 
> Too bad the sequence fanatics are scattered over
> the whole world, otherwise we could have a party.
> Email can't do everything!
> 
> Thanks, Marc - and best wishes to everyone for 2002
> 
> Neil

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?




From mlb at well.com  Wed Dec 12 21:49:45 2001
From: mlb at well.com (Marc LeBrun)
Date: Wed, 12 Dec 2001 12:49:45 -0800
Subject: Jeroboam
In-Reply-To: <200112121950.OAA33220@fry.research.att.com>
Message-ID: <5.1.0.14.2.20011212122511.00ab17b0@mail.well.com>

 >=N. J. A. Sloane
 > Too bad the sequence fanatics are scattered over the
 > whole world, otherwise we could have a party.

So we'll just have a distributed celebration.  Maybe convene a conference 
for A131072.  This will require a methuselah (although hopefully not a 
millennium!)

(More interesting numbers: "Methuselah was an antediluvian patriarch 
described in the Old Testament as having lived 969 years and whose name is 
synonymous with great age.  He may well have evolved from a character of 
earlier Sumerian legend who lived for 65,000 years.  To the Old Testament 
scribes this was perhaps too tall a tale, so they may have cut him back to 
a more conservative lifespan." --www.giantbottles.com)

Cheers!





From rpratt at email.unc.edu  Wed Dec 12 22:21:33 2001
From: rpratt at email.unc.edu (Rob Pratt)
Date: Wed, 12 Dec 2001 16:21:33 -0500 (EST)
Subject: antediluvian
In-Reply-To: <5.1.0.14.2.20011212122511.00ab17b0@mail.well.com>
Message-ID: <Pine.A41.4.21L1.0112121618390.50880-100000@login6.isis.unc.edu>

On Wed, 12 Dec 2001, Marc LeBrun wrote:

> ...
> 
> (More interesting numbers: "Methuselah was an antediluvian patriarch 
> described in the Old Testament as having lived 969 years and whose name is 
> synonymous with great age.  He may well have evolved from a character of 
> earlier Sumerian legend who lived for 65,000 years.  To the Old Testament 
> scribes this was perhaps too tall a tale, so they may have cut him back to 
> a more conservative lifespan." --www.giantbottles.com)
> 
> Cheers!

Note that Methuselah, Noah's grandfather, died in the year of The Flood
but wasn't on the ark.

Rob Pratt
Department of Operations Research
The University of North Carolina at Chapel Hill

rpratt at email.unc.edu

http://www.unc.edu/~rpratt/






From wilson at aprisma.com  Wed Dec 12 23:10:36 2001
From: wilson at aprisma.com (David W. Wilson)
Date: Wed, 12 Dec 2001 17:10:36 -0500
Subject: sumtimes product dividing n
References: <Pine.A41.4.33.0112121217190.31460-100000@gpu1.srv.ualberta.ca>
Message-ID: <3C17D5DC.EDE84D66@aprisma.com>

"M.V Subbarao" wrote:
> 
> I wonder if there are infinitely many integers n such that the sum of its
> digits times product of its digits is a divisor of n. On  example is n =
> 12. No doubt, we  can easily find many more.
> 
> M.V.Subbarao
> m.v.subbarao at ualberta.ca

Empirically, it looks as if every number of the form (10^3^n-1)/9 has the
desired property, which would make A049102 infinite.  Needs a proof, tho.




From mlb at well.com  Thu Dec 13 08:22:02 2001
From: mlb at well.com (Marc LeBrun)
Date: Wed, 12 Dec 2001 23:22:02 -0800
Subject: yet another unexpected EIS hit
Message-ID: <5.1.0.14.2.20011212223937.0408c178@mail.well.com>

Here's a pretty pair of dots that someone might be able to connect:

Define a "numbral arithmetic" by replacing addition with binary bitwise 
inclusive-OR (so that [3] + [5] = [7] etc) and multiplication becomes 
shift-&-OR instead of shift-&-add (so that [3] * [3] = [7] etc).

These numbrals have some interesting interpretations (such as a kind of 
"infinite base" system, or alternatively as sets of integers) that for 
brevity I'll resist  presenting.

Anyway, we can say naturally that [d] divides [n] when there exists an [e] 
such that

   [d] * [e] = [n].

For example the six divisors of [14] are [1], [2], [3], [6], [7] and [14].

Dot X: Counting the proper divisors of [2^n-1] gives the sequence

   0, 1, 2, 4, 7, 13, 23, 42, 76, 139, 255, 471, 873, 1627, 3044, 5718, ...

Dot Y: But this appears to be A048888, the anti-diagonal sums of table 
A048887 (qv)

            1  1  1  1  1  1  1 ...
            1  2  3  5  8 13 ...
            1  2  4  7 13 ...
            1  2  4  8 ...
            ...

where A(i,j) is the number of compositions of j into parts all <=i.

?!





From jawbrey at oakland.edu  Thu Dec 13 18:06:02 2001
From: jawbrey at oakland.edu (Jon Awbrey)
Date: Thu, 13 Dec 2001 12:06:02 -0500
Subject: lattice love song
Message-ID: <3C18DFFA.EF4AB088@oakland.edu>

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

seq fantasians,

did not know if folks had seen/heard this:

http://www.vub.ac.be/CLEA/aerts/latticelovesong.html

jon awbrey

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?




From njas at research.att.com  Sun Dec 16 06:39:02 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Sun, 16 Dec 2001 00:39:02 -0500 (EST)
Subject: report
Message-ID: <200112160539.AAA15347@fry.research.att.com>

There is a new version of the index, 
http://www.research.att.com/~njas/sequences/Sindx.html
which should be much faster

This involved many manual edits, so let me know if there are
errors or bad links

There will probably not be any further updates of the database
after tomorrow until the end of the year as I will be traveling
and unable to read email.  In case of emergency leave me voicemail
at (732) 828 6098.

Best wishes for the New Year!

Neil




From jawbrey at oakland.edu  Sun Dec 16 07:01:17 2001
From: jawbrey at oakland.edu (Jon Awbrey)
Date: Sun, 16 Dec 2001 01:01:17 -0500
Subject: report
References: <200112160539.AAA15347@fry.research.att.com>
Message-ID: <3C1C38AD.5DA2F17D@oakland.edu>

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

| Wollust ward dem Worm gegeben,
| Und der Cherub steht vor Gott!
|
| Friedrich von Schiller, Ode "An die Freude"

happy beethoven's birthday
and have a joyful new year!

jon awbrey

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

N. J. A. Sloane wrote:
> 
> There is a new version of the index,
> http://www.research.att.com/~njas/sequences/Sindx.html
> which should be much faster
> 
> This involved many manual edits, so let me know if there are
> errors or bad links
> 
> There will probably not be any further updates of the database
> after tomorrow until the end of the year as I will be traveling
> and unable to read email.  In case of emergency leave me voicemail
> at (732) 828 6098.
> 
> Best wishes for the New Year!
> 
> Neil

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?




From amarnath_murthy at yahoo.com  Sun Dec 16 08:18:23 2001
From: amarnath_murthy at yahoo.com (murthy amarnath)
Date: Sat, 15 Dec 2001 23:18:23 -0800 (PST)
Subject: report
In-Reply-To: <200112160539.AAA15347@fry.research.att.com>
Message-ID: <20011216071823.87356.qmail@web9604.mail.yahoo.com>


--- "N. J. A. Sloane" <njas at research.att.com> wrote:
> There is a new version of the index, 
>
http://www.research.att.com/~njas/sequences/Sindx.html
> which should be much faster
> 
> This involved many manual edits, so let me know if
> there are
> errors or bad links
> 
> There will probably not be any further updates of
> the database
> after tomorrow until the end of the year as I will
> be traveling
> and unable to read email.  In case of emergency
> leave me voicemail
> at (732) 828 6098.
> 
> Best wishes for the New Year!
> 
> Neil

Merry christmas and happy new year.
rgds
amarnath murthy


__________________________________________________
Do You Yahoo!?
Check out Yahoo! Shopping and Yahoo! Auctions for all of
your unique holiday gifts! Buy at http://shopping.yahoo.com
or bid at http://auctions.yahoo.com




From karttu at megabaud.fi  Mon Dec 17 19:00:32 2001
From: karttu at megabaud.fi (Antti Karttunen)
Date: Mon, 17 Dec 2001 20:00:32 +0200
Subject: An article in Frankfurter Allgemeine Zeitung
Message-ID: <3C1E32C0.E77F892E@megabaud.fi>


Jon Awbrey wrote:

  ?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

  | Wollust ward dem Worm gegeben,
  | Und der Cherub steht vor Gott!
  |
  | Friedrich von Schiller, Ode "An die Freude"

  happy beethoven's birthday
  and have a joyful new year!


Warm Christmas greetings from Finland also!


To delight your souls, I offer the article that appeared in 
Nr. 107 of Frankfurter Allgemeine Zeitung (09.05.2001),
as faithfully translated to English by
http://www.google.com/language_tools

To see the original Deutsch version, read:
http://www.megabaud.fi/~karttu/matikka/sloane.html

and this "English" version is also available as:
http://www.megabaud.fi/~karttu/matikka/sloane_eng.html

(I fetched the article from the electronic archives of "Frankfurter
Allgemeine"
http://afaz.gbi.de/ (and choose "Suche", then enter "Sloane" to "Suche"
field)
and it cost me just 1,50 euros, that is, a bit over one dollar).


Terveisin,

Antti Karttunen



   Nature and science Frankfurt general newspaper, 09,05,2001, NR. 107,
S. N1 
  
__________________________________________________________________________
                                                                         
   The passion of a number row collecting tank

   Data base with more than 61,000 entries/interest with mathematicians
and
   laymen largely 

   Some humans collect stamps, other coins, calling cards, beer covers
   or butterflies. There are hardly something, which did not become the
   object of human collecting passion, even night pots and tying lacings
   its lovers found. But for the probably most unusual collecting
objects
   the American mathematician Neil J. A has himself. Sloane of the AT&T
   Shannon lab in Florham park/new jersey decided. He collects
zahlenreihen.
   However not any arbitrary, but only such, which consist many elements
of
   positive whole numbers, are infinitely have and in addition according
to
   a firm rule developed.
  
   Although Sloane is probably the only collecting tank of zahlenreihen
in
   the world, its hobby encounters a broad interest. Thousands of
scientists
   and laymen help him for many years to constantly extend its
collection.
   In December 1963 Sloane, which was at this time still a student to
the
   Cornell University in Ithaca/New York, looked for information about a
   certain zahlenreihe from the graph theory. But as it also strove
itself,
   it could find nothing over it in the relevant literature. That
annoyed
   it so much that he began to collect systematically zahlenreihen.
 
   Later its collection over 2300 rows from all ranges of mathematics,
   the natural sciences and even the mental exercise covered ten years.
   It arranged it lexically and published it as book with the title
   "A Handbook OF Integer Sequences". The book became a success, and
many
   humans sent to it thereupon new rows. Neil Sloane continued to
collect.
   it wrote 1995 together with Simon Plouffe of the Universit? you
Qu?bec in
   Montr?al the "Encyclopedia OF Integer Sequences", which was than
twice
   as large with 5488 zahlenreihen more like its first collection.

   In the same year Sloane furnished E-Mail addresses, with which one
could
   make autopollings to its number row collection. The book and the
E-Mail
   addresses were a large success and led to an enormous tide of entries
   with new rows. One year later had already increased the collection on
   16,000 rows. Now Sloane arranged also its own InterNet side for its
   number row collection with special search functions
   ( http://www.research.att.com/~njas/sequences/ ).
   The interest among scientists and also among laymen is enormous. Per
day
   for instance 2500mal his collection one accesses, which contains in
the
   meantime over 61,000 rows.

   The collection of Sloane resembles a well sorted department store.
   All only somehow conceivable zahlenreihen are to be found there.
   Mathematical rows like those of the prime numbers (2, 3, 5, 7,
11...),
   the quadratzahlen (0, 1, 4, 9, 16...) or the faculties (1, 1, 2, 6,
24...)
   are naturally numerously represented. In addition, Neil Sloane seized
   numbers of chemistry like the number of the different alkanes with
   n carbon atoms (1, 1, 1, 2, 3, 5...) or numbers of physics like the
   number of the Feynman graphs of the order 2n (1, 3, 18, 153, 1638...)
   as well as numbers of biology like the possible secondary structures
   of a RNA molecule with n nucleotides (1, 1, 1, 2, 4, 8, 17...).

   In addition the collection contains chess problems like the number of
   the possibilities of placing n ladies in such a way on a chessboard
   with n fields that they do not threaten themselves mutually
   (1, 0, 0, 2, 10, 4, 40, 92...). To find curiosities are additional
   like the row 0, 0, 0, 0, 4, 9, 5, 1, 1, 0, 55... It results from the
   fact that one paints three, four, from the English number words one,
   two, five... all letters up to the number characters I, V, X, L, C, D
   and M. The word remainders are then interpreted as Roman numbers.

   In mystery columns and with intelligence tests question
   "as, liked, is called the next number?" a given zahlenreihe is easy
   with Sloanes collection to solve. If one enters the row 3, 1, 4, 1, 5
   for example into the search program, it offers to thirty-five
different
   possibilities, how the row could continue. One of the resulting rows
   would be from sequential whole numbers - beginning with the three -,
   which are separate by ones in each case. The next number would have
   to be thus unity. It could concern in addition, the decimal places of
the
   circle number of pi. Then the next element would have to be nine.
   Since 1998 the American mathematician even gives a special electronic
   magazine, which "journal OF Integer Sequences", out, in which
excluding
   articles over zahlenreihen appear.

   HEINRICH RESTRAIN

   All rights reserve. (C) F.A.Z . GmbH, Frankfurt/Main




From reinermartin at nyc.rr.com  Tue Dec 18 05:22:36 2001
From: reinermartin at nyc.rr.com (Reiner Martin)
Date: Mon, 17 Dec 2001 23:22:36 -0500
Subject: An article in Frankfurter Allgemeine Zeitung
References: <3C1E32C0.E77F892E@megabaud.fi>
Message-ID: <00e301c1877b$a2646460$6e7ba8c0@nyc.rr.com>


Since the automatic translation was a bit crude, I tried to translate this
article
myself as good as I could (my mother tongue is German). Here it is:

------------------------------------

Frankfurter Allgemeine Zeitung, May 9, 2001
Section: Nature and Science
Title: The Passion of a Integer Sequence Collector
Subtitle: Database with more than 61,000 entries / Wide interest with
mathematicians and amateurs


Some people collect stamps, other coins, calling cards, beer mats or
butterflies. There is hardly something which has not become the object of
the human passion to collect, even chamber pots and shoe laces have found
their devotees.  But the American mathematician Neil J. A. Sloane of the
AT&T Shannon Lab in Florham Park/New Jersey has probably chosen the most
unusual objects to collect.  He collects integer sequences.  Not any
arbitrary sequences however, but only such which consist of positive
integers, which have infinitely many elements, and which are build according
to a fixed rule.

Although Sloane is probably the only collector of integer sequences in the
world, his hobby is met with wide interest.  Thousands of scientists and
amateurs are helping him for many years now to continuously extend his
collection.  In December 1963 Sloane, who was at this time still a student
of Cornell University in Ithaca/New York, looked for information about a
certain sequence from graph theory.  But as hard as he tried, he could not
find anything about it in the relevant literature.  That annoyed him so much
that he began to collect sequences systematically.

Ten years later his collection contains over 2300 sequences from all areas
of mathematics, the natural sciences and even from puzzles.  He arranged
them lexically and published them as book with the title "A Handbook of
Integer Sequences."  The book became a success, and many people sent him new
sequences.  Neil Sloane continued to collect.  Together with Simon Plouffe
of the Universit? du Qu?bec in Montr?al he wrote in 1995 the "Encyclopedia
of Integer Sequences", which was with 5488 sequences more than twice as
large as his first collection.

In the same year Sloane created e-mail addresses with which one could make
automatic look-ups in his sequence database.  The book and the e-mail
addresses were a large success and led to an enormous wave of contributions
of new sequences.  One year later the collection had already increased to
16,000 sequences.  Then Sloane created also an Internet page for his
sequence database with special search functions
(www.research.att.com/~njas/sequences).  The interest among scientists and
also among amateurs is enormous.  Every day his collection is accessed about
2,500 times, which in the meantime contains over 61,000 sequences.

The collection of Sloane resembles a well sorted department store.  All
somehow conceivable sequences are to be found there.  Mathematical sequences
like those of the prime numbers (2, 3, 5, 7, 11 ...), the square numbers (0,
1, 4, 9, 16 ...) or the factorials (1, 1, 2, 6, 24 ...) are of course
represented numerously.  In addition, Neil Sloane added sequences from
chemistry like the number of different alkanes with n carbon atoms (1, 1, 1,
2, 3, 5 ...), or sequences from physics like the number of Feynman diagrams
of order 2n (1, 3, 18, 153, 1638...), as well as sequences from biology like
the possible secondary structures of a RNA molecule with n nucleotides (1,
1, 1, 2, 4, 8, 17 ...).

Additionally, the collection contains chess problems like the number of ways
of placing n queens on a chessboard with n by n squares in such a way that
they do not mutually attack themselves (1, 0, 0, 2, 10, 4, 40, 92 ...).  One
can also find curiosities like the sequence 0, 0, 0, 0, 4, 9, 5, 1, 1, 0, 55
...  It results from removing all letters except the number characters I, V,
X, L, C, D and M from the English numbers one, two, three, four, five, ...
The resulting words are then interpreted as Roman numbers.

The question "what is the next number?" of a given sequence, which is
popular in puzzle columns and intelligence tests, is easy to solve with
Sloanes collection.  For example, if one enters the sequence 3, 1, 4, 1, 5
into the search program, it offers thirty-five different possibilities to
continue the sequence.  One of the resulting sequences would be the one of
all integers, beginning with three, separated by ones.  Thus, the next
number would have to be one.  But it could also represent the decimals of
the number pi.  Then the next element would have to be nine.  Since 1998 the
American mathematician even publishes a special electronic magazine, the
"Journal of Integer Sequences", in contains exclusively articles on integer
sequences.

HEINRICH HEMME

Translated from the German by Reiner Martin

All rights reserved.  (C)  F.A.Z .  GmbH, Frankfurt/Main



----- Original Message -----
From: "Antti Karttunen" <karttu at megabaud.fi>
To: <seqfan at ext.jussieu.fr>
Sent: Monday, December 17, 2001 1:00 PM
Subject: An article in Frankfurter Allgemeine Zeitung


>
> Jon Awbrey wrote:
>
>   ?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?
>
>   | Wollust ward dem Worm gegeben,
>   | Und der Cherub steht vor Gott!
>   |
>   | Friedrich von Schiller, Ode "An die Freude"
>
>   happy beethoven's birthday
>   and have a joyful new year!
>
>
> Warm Christmas greetings from Finland also!
>
>
> To delight your souls, I offer the article that appeared in
> Nr. 107 of Frankfurter Allgemeine Zeitung (09.05.2001),
> as faithfully translated to English by
> http://www.google.com/language_tools
>
> To see the original Deutsch version, read:
> http://www.megabaud.fi/~karttu/matikka/sloane.html
>
> and this "English" version is also available as:
> http://www.megabaud.fi/~karttu/matikka/sloane_eng.html
>
> (I fetched the article from the electronic archives of "Frankfurter
> Allgemeine"
> http://afaz.gbi.de/ (and choose "Suche", then enter "Sloane" to "Suche"
> field)
> and it cost me just 1,50 euros, that is, a bit over one dollar).
>
>
> Terveisin,
>
> Antti Karttunen
>
>
>
>    Nature and science Frankfurt general newspaper, 09,05,2001, NR. 107,
> S. N1
>
> __________________________________________________________________________
>
>    The passion of a number row collecting tank
>
>    Data base with more than 61,000 entries/interest with mathematicians
> and
>    laymen largely
>
>    Some humans collect stamps, other coins, calling cards, beer covers
>    or butterflies. There are hardly something, which did not become the
>    object of human collecting passion, even night pots and tying lacings
>    its lovers found. But for the probably most unusual collecting
> objects
>    the American mathematician Neil J. A has himself. Sloane of the AT&T
>    Shannon lab in Florham park/new jersey decided. He collects
> zahlenreihen.
>    However not any arbitrary, but only such, which consist many elements
> of
>    positive whole numbers, are infinitely have and in addition according
> to
>    a firm rule developed.
>
>    Although Sloane is probably the only collecting tank of zahlenreihen
> in
>    the world, its hobby encounters a broad interest. Thousands of
> scientists
>    and laymen help him for many years to constantly extend its
> collection.
>    In December 1963 Sloane, which was at this time still a student to
> the
>    Cornell University in Ithaca/New York, looked for information about a
>    certain zahlenreihe from the graph theory. But as it also strove
> itself,
>    it could find nothing over it in the relevant literature. That
> annoyed
>    it so much that he began to collect systematically zahlenreihen.
>
>    Later its collection over 2300 rows from all ranges of mathematics,
>    the natural sciences and even the mental exercise covered ten years.
>    It arranged it lexically and published it as book with the title
>    "A Handbook OF Integer Sequences". The book became a success, and
> many
>    humans sent to it thereupon new rows. Neil Sloane continued to
> collect.
>    it wrote 1995 together with Simon Plouffe of the Universit? you
> Qu?bec in
>    Montr?al the "Encyclopedia OF Integer Sequences", which was than
> twice
>    as large with 5488 zahlenreihen more like its first collection.
>
>    In the same year Sloane furnished E-Mail addresses, with which one
> could
>    make autopollings to its number row collection. The book and the
> E-Mail
>    addresses were a large success and led to an enormous tide of entries
>    with new rows. One year later had already increased the collection on
>    16,000 rows. Now Sloane arranged also its own InterNet side for its
>    number row collection with special search functions
>    ( http://www.research.att.com/~njas/sequences/ ).
>    The interest among scientists and also among laymen is enormous. Per
> day
>    for instance 2500mal his collection one accesses, which contains in
> the
>    meantime over 61,000 rows.
>
>    The collection of Sloane resembles a well sorted department store.
>    All only somehow conceivable zahlenreihen are to be found there.
>    Mathematical rows like those of the prime numbers (2, 3, 5, 7,
> 11...),
>    the quadratzahlen (0, 1, 4, 9, 16...) or the faculties (1, 1, 2, 6,
> 24...)
>    are naturally numerously represented. In addition, Neil Sloane seized
>    numbers of chemistry like the number of the different alkanes with
>    n carbon atoms (1, 1, 1, 2, 3, 5...) or numbers of physics like the
>    number of the Feynman graphs of the order 2n (1, 3, 18, 153, 1638...)
>    as well as numbers of biology like the possible secondary structures
>    of a RNA molecule with n nucleotides (1, 1, 1, 2, 4, 8, 17...).
>
>    In addition the collection contains chess problems like the number of
>    the possibilities of placing n ladies in such a way on a chessboard
>    with n fields that they do not threaten themselves mutually
>    (1, 0, 0, 2, 10, 4, 40, 92...). To find curiosities are additional
>    like the row 0, 0, 0, 0, 4, 9, 5, 1, 1, 0, 55... It results from the
>    fact that one paints three, four, from the English number words one,
>    two, five... all letters up to the number characters I, V, X, L, C, D
>    and M. The word remainders are then interpreted as Roman numbers.
>
>    In mystery columns and with intelligence tests question
>    "as, liked, is called the next number?" a given zahlenreihe is easy
>    with Sloanes collection to solve. If one enters the row 3, 1, 4, 1, 5
>    for example into the search program, it offers to thirty-five
> different
>    possibilities, how the row could continue. One of the resulting rows
>    would be from sequential whole numbers - beginning with the three -,
>    which are separate by ones in each case. The next number would have
>    to be thus unity. It could concern in addition, the decimal places of
> the
>    circle number of pi. Then the next element would have to be nine.
>    Since 1998 the American mathematician even gives a special electronic
>    magazine, which "journal OF Integer Sequences", out, in which
> excluding
>    articles over zahlenreihen appear.
>
>    HEINRICH RESTRAIN
>
>    All rights reserve. (C) F.A.Z . GmbH, Frankfurt/Main






From karttu at megabaud.fi  Tue Dec 18 10:33:03 2001
From: karttu at megabaud.fi (Antti Karttunen)
Date: Tue, 18 Dec 2001 11:33:03 +0200
Subject: An article in Frankfurter Allgemeine Zeitung
References: <3C1E32C0.E77F892E@megabaud.fi>
    <00e301c1877b$a2646460$6e7ba8c0@nyc.rr.com>
Message-ID: <3C1F0D4F.C96EB42A@megabaud.fi>



Reiner Martin wrote:

> Since the automatic translation was a bit crude, I tried to translate this
> article
> myself as good as I could (my mother tongue is German).

Thanks!
I just tested the translation also with Altavista's Babelfish
at http://world.altavista.com/tr
and realized that its German-to-English translation almost
certainly uses the same software as Google at
http://www.google.com/language_tools
but with a slightly differing lexicon.


Terveisin,

Antti Karttunen






From njas at research.att.com  Tue Dec 18 15:11:05 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Tue, 18 Dec 2001 09:11:05 -0500 (EST)
Subject: FAZ
Message-ID: <200112181411.JAA65939@fry.research.att.com>

Thanks very much to  Antti Karttunen for the German version
and to Reiner Martin for the excellent translation of the F.A.Z. article.

I made a few small changes to the English version and put both versions
on the "Welcome to the OEIS" page (Seis.html) (with Reiner's permission)

NJAS




From layman at calvin.math.vt.edu  Tue Dec 18 19:11:31 2001
From: layman at calvin.math.vt.edu (John Layman)
Date: Tue, 18 Dec 2001 13:11:31 -0500 (EST)
Subject: More on "numbral" divisors
Message-ID: <20011218181145Z10637-16465+28@calvin.math.vt.edu>

On Dec 12, 2001, Marc LeBrun (mlb at mail.well.com) conjectured that 
the number of proper "numbral" divisors of 2^n-1 gives A048888, 
later confirmed by Richard Schroeppel (rcs at cs.arizona.edu).  
Calculations that I have made recently suggest the following possibly 
related conjecture concerning A007059 (balanced ordered trees with 
n nodes).

Conjecture: For n>=1, A007059(n+1) is the number of "numbral" divisors 
of (4^n-1)/3 = A002450(n).

Can anyone confirm this?

Further, I have shown that A002450(n) = (4^n-1)/3 is the nth umbral 
power of 5.





From karttu at megabaud.fi  Wed Dec 19 13:35:38 2001
From: karttu at megabaud.fi (Antti Karttunen)
Date: Wed, 19 Dec 2001 14:35:38 +0200
Subject: More on "numbral" divisors
References: <20011218181145Z10637-16465+28@calvin.math.vt.edu>
Message-ID: <3C20899A.5C80C901@megabaud.fi>



John Layman wrote:

> On Dec 12, 2001, Marc LeBrun (mlb at mail.well.com) conjectured that
> the number of proper "numbral" divisors of 2^n-1 gives A048888,
> later confirmed by Richard Schroeppel (rcs at cs.arizona.edu).

Dear John, Richard, others.

A little wish: Could you publish these little proofs somewhere
(e.g. as an associated notes file stored under EIS for the corresponding
sequence)?
I guess "HAKMEM" http://www.inwap.com/pdp10/hbaker/hakmem/hakmem.html
style would be fine.
Unless of course the proof is longer, and will be published in
a journal some day.

I think the value of EIS increases a year by year (contrary to
some other opinions...), especially when the important
sequences contain more and more references to the literature,
to the existing quality web sites (containing proofs also), and
to the other related EIS-sequences of the importance.



Merry christmas,

Antti Karttunen






From ogerard at ext.jussieu.fr  Wed Dec 19 15:34:45 2001
From: ogerard at ext.jussieu.fr (Olivier Gerard)
Date: Wed, 19 Dec 2001 15:34:45 +0100
Subject: Seqfan Site (was Re: More on "numbral" divisors)
In-Reply-To: <3C20899A.5C80C901@megabaud.fi>
References: <20011218181145Z10637-16465+28@calvin.math.vt.edu>
    <3C20899A.5C80C901@megabaud.fi>
Message-ID: <20011219143445.GD16003@ibazardev.ibazar-group.com>


To all seqfan members,

the site seqfan.net can be used by any EIS contributor to
store permanently any material related to sequences.

Material can be indexed by sequence number for urls

(ex http://www.seqfan.net/Axxxxxx/   )

or by a contributor log name.
As I can use symbolic links, the same material
can be attributed to several sequences.

Once more than one page is available I will make
two index pages available from the home page
for looking up material.

CGI/dynamic pages can be written in perl and php.

Just mail it to me for inclusion (please put Neil
in copy each time).

Olivier


On Wed, Dec 19, 2001 at 02:35:38PM +0200, Antti Karttunen wrote:
> 
> Dear John, Richard, others.
> 
> A little wish: Could you publish these little proofs somewhere
> (e.g. as an associated notes file stored under EIS for the corresponding
> sequence)?
> I guess "HAKMEM" http://www.inwap.com/pdp10/hbaker/hakmem/hakmem.html
> style would be fine.
> Unless of course the proof is longer, and will be published in
> a journal some day.
> 
> I think the value of EIS increases a year by year (contrary to
> some other opinions...), especially when the important
> sequences contain more and more references to the literature,
> to the existing quality web sites (containing proofs also), and
> to the other related EIS-sequences of the importance.
> 
> 
> 
> Merry christmas,
> 
> Antti Karttunen
> 
> 
> 




From karttu at megabaud.fi  Wed Dec 19 17:43:43 2001
From: karttu at megabaud.fi (Antti Karttunen)
Date: Wed, 19 Dec 2001 18:43:43 +0200
Subject: Seqfan Site
References: <20011218181145Z10637-16465+28@calvin.math.vt.edu>
    <3C20899A.5C80C901@megabaud.fi>
    <20011219143445.GD16003@ibazardev.ibazar-group.com>
Message-ID: <3C20C3BE.3D09F90B@megabaud.fi>


Dear Olivier,

seeing that there's not yet much activity in www.seqfan.net,
I suggest that you make the archives of SeqFan-list
(from  Listserv at Ext.jussieu.fr) available there
(either as a static Web-pages, or through some dynamic
script), because currently they are very difficult to access
via SeqFan-bitserv's own request-mechanism,
unless you want to delimit access only to SeqFan-members
themselves. Indeed, the publishing of e-mail addresses
occurring in the headers of SeqFan-messages
to any web-browsing robot might be a problem viz-a-viz spammers.
On the other hand, I guess that the most SeqFanatics
that have mailed to the list have already published their
mail-addresses in the entries they have submitted to EIS.

I guess the voice of the SeqFan-members should
be heard on this... Maybe one could use a "clever"
anti-spamming substitution like
sed -e 's/@/@supprimez./g' when transferring
the archives in bulk..
(This probably breaks some Mathematica code also,
so should preferably be applied only to the headers
and nearby of the messages.)

Salut,

  Antti


Olivier Gerard wrote:

> To all seqfan members,
>
> the site seqfan.net can be used by any EIS contributor to
> store permanently any material related to sequences.
>
> Material can be indexed by sequence number for urls
>
> (ex http://www.seqfan.net/Axxxxxx/   )
>
> or by a contributor log name.
> As I can use symbolic links, the same material
> can be attributed to several sequences.
>
> Once more than one page is available I will make
> two index pages available from the home page
> for looking up material.
>
> CGI/dynamic pages can be written in perl and php.
>
> Just mail it to me for inclusion (please put Neil
> in copy each time).
>
> Olivier
>





From frank.ellermann at t-online.de  Wed Dec 19 18:33:46 2001
From: frank.ellermann at t-online.de (Frank Ellermann)
Date: Wed, 19 Dec 2001 18:33:46 +0100
Subject: Seqfan Site
References: <20011218181145Z10637-16465+28@calvin.math.vt.edu>
    <3C20899A.5C80C901@megabaud.fi>
    <20011219143445.GD16003@ibazardev.ibazar-group.com>
    <3C20C3BE.3D09F90B@megabaud.fi>
Message-ID: <3C20CF7A.4DCA@t-online.de>

Hello Antti, you wrote:

> Indeed, the publishing of e-mail addresses
> occurring in the headers of SeqFan-messages
> to any web-browsing robot might be a problem
...
> I guess the voice of the SeqFan-members should
> be heard on this...

If the SeqFan archives are completely public, then
we could also organize it as newsgroup and retrieve
old stuff with Google.  Or in other words, I don't
like THIS idea, because I want to post really stupid
questions in SeqFan more privately, and not ready to 
be read by Google users in 2121... ;-)

But I like your original idea:  Collecting infos for
EIS sequences, especially sources used to calculate
terms in EIS sequences.  Probably you won't like my
REXX scripts like I can't use PARI, MAPLE, etc. here
directly, but often _any_ source is better than no
source at all.  Until now long programs are not shown
as %p in EIS, maybe seqfan.net files could solve this
problem ?

Example: Somebody computed A035615(26) this year, but
I don't know how.  All I have is my own algorithm and
some ideas, which needs about a day on my system to
verify A035615(16): yes, it's correct, but I want an
algorithm for more terms and not less... ;-)

                   Bye, Frank





From njas at research.att.com  Wed Dec 19 18:57:44 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Wed, 19 Dec 2001 12:57:44 -0500 (EST)
Subject: Seqfan Site
Message-ID: <200112191757.MAA34726@fry.research.att.com>


I am also perfectly willing to include plain text files 
on the sequence database web site, with links to them
from the apropriate entries.

There are already quite a few of these.  See for example

%H A005574 F. Ellermann, <a href="http://www.research.att.com/~njas/sequences/a002496.txt">Primes of the form (m^2)+1 up to 10^6</a>

%H A061396 V. Jovovic, <a href="http://www.research.att.com/~njas/sequences/a061396.txt">First 100 terms</a>

%H A006968 Gerard Schildberger, <a href="http://www.research.att.com/~njas/sequences/a006968.txt">The first 3999 numbers in Roman numerals</a>

%H A066057 K. Brockhaus, <a href="http://www.research.att.com/~njas/sequences/a058042.txt">On the'Reverse and Add!' algorithm in base 2</a>

etc

NJAS




From ogerard at ext.jussieu.fr  Wed Dec 19 20:26:28 2001
From: ogerard at ext.jussieu.fr (Olivier Gerard)
Date: Wed, 19 Dec 2001 20:26:28 +0100
Subject: Seqfan Site, Newsgroup, etc.
In-Reply-To: <3C20C3BE.3D09F90B@megabaud.fi>
References: <20011218181145Z10637-16465+28@calvin.math.vt.edu>
    <3C20899A.5C80C901@megabaud.fi>
    <20011219143445.GD16003@ibazardev.ibazar-group.com>
    <3C20C3BE.3D09F90B@megabaud.fi>
Message-ID: <20011219192628.GF16003@ibazardev.ibazar-group.com>


Dear Antti and all,

I have reserved and used as a front page the domain name seqfan.net
to have a place in common for us for *public* information about
seqfan but also sequences.

I welcome any idea on making this a better companion site to the EIS.
As I said before, I have large control over what can be put there,
and not only text files.

The mailing list is another thing altogether.  My already long
experience on newsgroups and mailing lists convinces me that
this list should stay private (i.e. no external posting allowed)
and as many of the members have subscribed knowing this was the
case, I have certainly no intent to post publicly any seqfan list
mail or email address without prior consent or initiative of the
members involved.

Now, the mail order archive system is cumbersome and without
visibility, especially for new members so I have prepared a private
browsable archive with monharc and a few other tools but requests
have been very seldom, and almost always from people able to
massage the archive in their own prefered format and tools.
So I have not put it online but I could if there is sufficient interest.
In this case I would make it restricted to seqfan members
so email harvesting would not be a concern.

We could consider making a newsgroup out of seqfan, but frankly
I think it would be a waste of time and peace.  Seqfan is already
very open (compare that to mathfun for the people who can), and
I never refuse someone who has made a contribution on the EIS.
My only requirement these days is to have a valid, non advertisement
postscripting email address and just to ask for it. This insures
the minimum level of motivation and interest to keep administrating
this list to a minimum compatable with my personal, academic and 
private life.

Like in a newsgroups, most members just read and don't contribute
but unlike most newsgroups, we don't need moderators and we don't 
experience flame wars.  The OEIS pages serve as a reference and
as a FAQ.  I am very much concerned about keeping the noise level
of seqfan as low as possible, and this not very easy in a newsgroup.

If you have reflexions on the newsgroup proposal or other modifications
of seqfan, please email them directly to Neil, me and Antti, because I don't
want to bother other members when possible on this meta-subject. 

Regards,

Olivier


On Wed, Dec 19, 2001 at 06:43:43PM +0200, Antti Karttunen wrote:
> 
> Dear Olivier,
> 
> seeing that there's not yet much activity in www.seqfan.net,
> I suggest that you make the archives of SeqFan-list
> (from  Listserv at Ext.jussieu.fr) available there
> (either as a static Web-pages, or through some dynamic
> script), because currently they are very difficult to access
> via SeqFan-bitserv's own request-mechanism,
> unless you want to delimit access only to SeqFan-members
> themselves. Indeed, the publishing of e-mail addresses
> occurring in the headers of SeqFan-messages
> to any web-browsing robot might be a problem viz-a-viz spammers.
> On the other hand, I guess that the most SeqFanatics
> that have mailed to the list have already published their
> mail-addresses in the entries they have submitted to EIS.
> 
> I guess the voice of the SeqFan-members should
> be heard on this... Maybe one could use a "clever"
> anti-spamming substitution like
> sed -e 's/@/@supprimez./g' when transferring
> the archives in bulk..
> (This probably breaks some Mathematica code also,
> so should preferably be applied only to the headers
> and nearby of the messages.)
> 
> Salut,
> 
>   Antti
> 




From mlb at well.com  Fri Dec 21 09:28:50 2001
From: mlb at well.com (Marc LeBrun)
Date: Fri, 21 Dec 2001 00:28:50 -0800
Subject: a tad more on OR-numbrals
Message-ID: <5.1.0.14.2.20011220231920.03c2ab48@mail.well.com>

A few tardy follow-on comments:

Many thanks to Rich Schroeppel for his clarifying explanation of the EIS 
hit.  I hope his "gapology" can be used to understand the full OR-numbral 
divisor sequence

   0 1 1 2 1 3 2 3 1 3 1 5 1 5 4 4 1 3 1 5 2 3 1 7 1 3 3 8 1 9 7 5 1 3 1 5 
1 3 1 7
   1 5 1 5 3 3 3 9 1 3 3 5 1 7 3 11 1 3 3 14 3 15 13 6 1 3 1 5 1 3 1 7 2 3 
1 5 1 3
   1 9 1 3 1 8 4 3 1 7 1 7 3 5 1 7 5 11 1 3 3 5 1 7 3 7 1 3 1 11 3 7 4 14 1 
3 3 5
   1 7 5 19 1 7 4...

which, along with a zillion others isn't in EIS yet because I can't fit the 
explanation in the "margin" (to address Antti Karttunen's complaint re. 
John Layman's interesting findings).  Maybe a numbral reference web page 
someday...

In the meantime, the two interpretations of OR-numbrals I alluded to are:

1. Sets of integers, with a 1 in the Nth bit denoting N's membership in the 
set.  Then numbral addition corresponds to set union, and multiplication 
means forming the set of all pair-wise sums.  (Maybe this would be useful 
for studying addition spectra or something).

2. Also, a homogeneous binary basis in powers of B, whose carries shift L 
places left, can be analyzed by solving
   B^N + B^N  =  B^(N+L)
to find the base B = 2^(1/L).  Thus the usual L=+1 gives vanilla binary 
B=2, L=-1 gives bit-reversed binary B=1/2, L=+2 gives the "tinker-toy" base 
B=sqrt(2), and so on.

When the carries don't shift at all addition degenerates into bitwise 
OR.  Here L=0, and we get B=2^(1/0).  So OR-numbrals are also a kind of 
"infinite" base.

But just what kind of "infinities" are these B^N=2^(N/0)?  Particularly 
that lsb "finity", 2^(0/0)?!

Can you unify these two interpretations?  A class of all sets of N things 
is somehow the same as some kind of Nth transfinite numeral?

(I suppose I should also mention that when you throw the carries away 
altogether you get XOR.  You might think of this as L=(minus) infinity, 
otherwise known as "polynomials over Z2" etc.  So perhaps either XOR is 
some kind of infinitesimal arithmetic, or else maybe a shaggy dog story 
about different flavors of zero...).

Each numbral system has its own "numbral theory" with analogs of 
partitions, divisors, etc that remain to be explored and sequenced.  If you 
come up with more such systems, interpretations and/or mysterious hits, 
please let me know.

Happy New Year!





From jens.voss at poet.de  Fri Dec 21 10:03:39 2001
From: jens.voss at poet.de (Jens Voss)
Date: Fri, 21 Dec 2001 10:03:39 +0100
Subject: a tad more on OR-numbrals
References: <5.1.0.14.2.20011220231920.03c2ab48@mail.well.com>
Message-ID: <3C22FAEB.29F85C96@poet.de>


Marc LeBrun wrote:
> 
> [...lots of interesting thoughts on different numbral systems]
> 

What strikes me most about the different numbral arithmetics is
that on one hand they are all nice since they are associative,
commutative and distributive, so the "divides" relation is
compatible with addition and multiplication. On the other hand,
the OR-numbral system appears to be the only one in which there
is no unique factorization, so as one consequence of that we have
to have to distinguish between "irreducible" and "prime" elements:

[5] for example is irreducible (since it is not a product of two
factors different from [1]), but not prime since it neither divides
[3] nor [11] but their product [3] * [11] = [31] = [5] * [7].

Gru?,
Jens

-- 
---------------------------------------------------------
Jens Voss, POET Software, Kattjahren 4 - 8, 22359 Hamburg
---------------------------------------------------------
The opinions expressed above are mine, not my employer's.
---------------------------------------------------------
"Tee-dah, tah-dee, tee-dah, tah-dee..."
                    J. Brahms, 4th symphony, 1st movement
---------------------------------------------------------




From njas at research.att.com  Sat Dec 22 19:09:38 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Sat, 22 Dec 2001 13:09:38 -0500 (EST)
Subject: small bug in %Y lines
Message-ID: <200112221809.NAA07269@fry.research.att.com>

There was a small bug in the program that processes new sequences and comments,
in the part that processes %Y or cross-reference lines.

The result was that in a few cases the whole %Y line was deleted.
-----------------------------------------------------------------

This only happened in the last two weeks, and did not
affect most submissions.

But if you kept a copy of recent submissions you might check to
see that the %Y lines were reproduced in the final version in the database.

The bug has now neen fixed.

I know it affected some recent submissions from Amarnath Murthy

Apologies

NJAS




From njas at research.att.com  Mon Dec 24 04:44:00 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Sun, 23 Dec 2001 22:44:00 -0500 (EST)
Subject: home for OEIS?
Message-ID: <200112240344.WAA82213@fry.research.att.com>

In view of recent developments at AT&T there is a possibility
that the On-Line Encyclopedia of Integer Sequences (and myself
- but that's another story) may need a new home one day.

One solution would be to get a domain name (OEIS.org, say) and
find an ISP to host it.  However, there are complications:
Size: over 256 MB
Operating system: all the lookup programs use Unix shell commands,
   superseeker also uses Maple, Mma, C, Fortran, etc
   so they would have to be available on the host machine.
Speed: needs a big fast Unix machine to get
   the rapid response we have now 

I have had no experience with ISP's.
If anyone has suggestions or advice please email me directly.
Neil Sloane, njas at research.att.com





From jawbrey at oakland.edu  Mon Dec 24 20:30:40 2001
From: jawbrey at oakland.edu (Jon Awbrey)
Date: Mon, 24 Dec 2001 14:30:40 -0500
Subject: Toward A Functional Conception Of Quantificational Logic
References: <3C21FA39.F189B7CC@oakland.edu>
    <3C220298.8D2147E4@oakland.edu> <3C251170.52243E10@oakland.edu>
    <3C255EAE.A4E5B20A@oakland.edu>
Message-ID: <3C278260.48418DBA@oakland.edu>

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

Note 128

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

Subj:  Toward A Functional Conception Of Quantificational Logic

I am going to put off explaining Table 11, that presents a sample of
what I call "Interpretive Categories for Higher Order Propositions",
until after we get beyond the 1-dimensional case, since these lower
dimensional cases tend to be a bit "condensed" or "degenerate" in
their structures, and a lot of what is going on here will almost
automatically become clearer as soon as we get even two logical
variables into the mix.

| Document History:
|
| Subject:  Inquiry & Analogy
| Contact:  Jon Awbrey <jawbrey at oakland.edu>
| Version:  Draft 3.21
| Created:  01 Jan 1995
| Revised:  24 Dec 2001
| Faculty:  F. Mili & M.A. Zohdy
| Setting:  Oakland University, Rochester, Michigan, USA
| Excerpt:  Section 2.1.2 (Higher Order Propositions & Logical Operators)

2.1.2  Higher Order Propositions & Logical Operators (n = 2)

By way of reviewing notation and preparing to extend it to
higher order universes of discourse, let us first consider
the universe of discourse X? = [$X$] = [x_1, x_2] = [x, y],
based on two logical features or boolean variables x and y.

1.  The points of X? are collected in the space:

    X   =   <<x, y>>   =   {<x, y>}   ~=~   %B%^2.

    In other words, written out in full:

    X   =   {<"(x)", "(y)">,
             <"(x)", " y ">,
             <" x ", "(y)">,
             <" x ", " y ">}

    X  ~=~  {<%0%, %0%>,
             <%0%, %1%>,
             <%1%, %0%>,
             <%1%, %1%>}

2.  The propositions of X? make up the space:

    ^X^  =  (X -> %B%)  =  {f : X -> %B%}  ~=~  (%B%^2 -> %B%).

As always, it is frequently convenient to omit a few of the
finer markings of distinctions among isomorphic structures,
so long as one is aware of their presence and knows when
it is crucial to call upon them again.

The next higher order universe of discourse that is built on X? is
X?2 = [X?] = [[x, y]], which may be developed in the following way.
The propositions of X? become the points of X?2, and the mappings
of the type m : (X -> %B%) -> %B% become the propositions of X?2.
In addition, it is convenient to equip the discussion with with
a selected set of higher order operators on propositions, all
of which have the form w : (%B%^2 -> %B%)^k -> %B%.

To save a few words in the remainder of this discussion, I will
use the terms "measure" and "qualifier" to refer to all types of
"higher order" (HO) propositions and operators.  To describe the
present setting in picturesque terms, the propositions of [x, y]
may be regarded as a gallery of sixteen venn diagrams, while the
measures m : (X -> %B%) -> %B% are analogous to a body of judges
or a collection of critical viewers, each of whom evaluates each
picture as a whole and reports the ones that find favor or not.
In this way, each judge m_j partitions the gallery of pictures
into two aesthetic portions, the pictures (m_j)^(-1)(%1%) that
m_j likes and the pictures (m_j)^(-1)(%0%) that m_j dislikes.

There are 2^16 = 65536 measures of type m : (%B%^2 -> %B%) -> %B%.
Table 12 introduces the first 16 of these measures in the fashion
of the HO truth table that I used before.  The column headed "m_j"
shows the values of the measure m_j on each of the propositions
f_i : %B%^2 -> %B%, for i = 0 to 15, with blank entries in the
Table being optional for values of zero.  The arrangement of
measures that continues according to the plan indicated here
will be referred to as the "standard ordering" of measures.
In this scheme of things, the index j of the measure m_j is
the decimal equivalent of the bit string that is associated
with m_j's functional values, which can be obtained in turn
by reading the j^th column of binary digits in the Table as
the corresponding range of boolean values, taking them up
in the order from bottom to top.

Table 12.  Higher Order Propositions (n = 2)
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
|  | x | 1100 |    f     |m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|.|
|  | y | 1010 |          |0|0|0|0|0|0|0|0|0|0|1|1|1|1|1|1|.|
| f \  |      |          |0|1|2|3|4|5|6|7|8|9|0|1|2|3|4|5|.|
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
|      |      |          |                                 |
| f_0  | 0000 |    ()    |0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1  |
|      |      |          |                                 |
| f_1  | 0001 |  (x)(y)  |    1 1 0 0 1 1 0 0 1 1 0 0 1 1  |
|      |      |          |                                 |
| f_2  | 0010 |  (x) y   |        1 1 1 1 0 0 0 0 1 1 1 1  |
|      |      |          |                                 |
| f_3  | 0011 |  (x)     |                1 1 1 1 1 1 1 1  |
|      |      |          |                                 |
| f_4  | 0100 |   x (y)  |                                 |
|      |      |          |                                 |
| f_5  | 0101 |     (y)  |                                 |
|      |      |          |                                 |
| f_6  | 0110 |  (x, y)  |                                 |
|      |      |          |                                 |
| f_7  | 0111 |  (x  y)  |                                 |
|      |      |          |                                 |
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
|      |      |          |                                 |
| f_8  | 1000 |   x  y   |                                 |
|      |      |          |                                 |
| f_9  | 1001 | ((x, y)) |                                 |
|      |      |          |                                 |
| f_10 | 1010 |      y   |                                 |
|      |      |          |                                 |
| f_11 | 1011 |  (x (y)) |                                 |
|      |      |          |                                 |
| f_12 | 1100 |   x      |                                 |
|      |      |          |                                 |
| f_13 | 1101 | ((x) y)  |                                 |
|      |      |          |                                 |
| f_14 | 1110 | ((x)(y)) |                                 |
|      |      |          |                                 |
| f_15 | 1111 |   (())   |                                 |
|      |      |          |                                 |
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o

HO, HO, HO, ...

Jon Awbrey

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?




From dp at dettodet.de  Thu Dec 27 11:09:14 2001
From: dp at dettodet.de (dp)
Date: Thu, 27 Dec 2001 11:09:14 +0100
Subject: A new LaTeX command: \anum{} and a comment
Message-ID: <MWMail.kfeacpce@host.none>



% ----------------------------------------------------------------
% LaTeX Paper *** anum.tex *** 2001-12-27
% Detlef Pauly, dp at dettodet.de
% **** -----------------------------------------------------------

\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage[colorlinks=true]{hyperref}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{8.9in}

% A new command: \anum{} -----------------------------------------

%with amsmath-package:
\newcommand{\anum}[1]
  {\text{\htmladdnormallink{A{#1}}
  {http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=#1}}}

%without amsmath-package:
%\newcommand{\anum}[1]
%  {\htmladdnormallink{A{#1}}
%  {http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=#1}}

% ----------------------------------------------------------------

\begin{document}

\textbf{A new \LaTeX\ command.}
\vspace{20pt}
\par\par
Hi seqfans.
\par
For example, to make a link to \anum{006125} use the small command

\begin{verbatim}
\anum{006125}
\end{verbatim}

instead of

\begin{footnotesize}
\begin{verbatim}
\htmladdnormallink{A006125}
{http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=006125}
\end{verbatim}
\end{footnotesize}

\par
Type 13 chars instead of 117 chars. Yep, $117 /13 = 9$.
\par

% ----------------------------------------------------------------

\vspace{30pt}
A new comment on \anum{006059} (Connected labeled topologies with $n$ points)
and \anum{001035} (Partially ordered sets [``posets''] with $n$ labeled elements
[or labeled acyclic transitive digraphs]):
%A006059(n)=n*A001035(n-1), n>0

\begin{equation*}
  \anum{006059}(n) = n \, \anum{001035}(n-1) \quad ,\text{for } n>0
\end{equation*}

\begin{small}
\anum{006059} 1, 1, 2, 9, 76, 1095, 25386, 910161, 49038872, 3885510411, \par
445110425110, 72721717736613, 16755380125270788, 5393244363726095487, 2405910197342218830914,
1477264863856923105482745, 1241074736327051013648799024, 1419169006353332682835352361843
\end{small}

% ----------------------------------------------------------------

\begin{table}[hb]
  \caption{Test}
  \label{tb:Test}
  \tabcolsep 4pt
  \renewcommand {\arraystretch} {1.3}
  \begin{small}
    \begin{tabular}{cccccccccc}
      \anum{000016} & \anum{000088} & \anum{000171} & \anum{000273} & \anum{000568} &
      \anum{000595} & \anum{000666} & \anum{000717} & \anum{000831} & \anum{001174} \\
      \anum{001187} & \anum{001349} & \anum{001437} & \anum{002499} & \anum{002500} &
      \anum{002785} & \anum{003027} & \anum{003030} & \anum{003085} & \anum{003086} \\
    \end{tabular}
  \end{small}
\end{table}

\par
Any comments/modifications?

\vspace{30pt}
ATB,
\par
DET \href{mailto:dp at dettodet.de}{dp at dettodet.de}

\end{document}

% ----------------------------------------------------------------





From Frederick.Magata at t-online.de  Sat Dec 29 07:17:09 2001
From: Frederick.Magata at t-online.de (Frederick Magata)
Date: Sat, 29 Dec 2001 07:17:09 +0100
Subject: 'reverse and add!'
Message-ID: <005101c19030$75764640$6e7ba8c0@ludwig>

Hello everyone,

this is my first contribution to the mailing list. So I hope you may find it
interesting.

In the context of the popular 'reverse and add!'-algorithm consider the
following sequence:
Let a(n) be the minimal number so that the 'reverse and add!'-algorithm in
base n does not terminate in a palindrome (in base n). If there is no such
number regarding base n, then a(n):=-1.

As proved by K. Brockhaus [1] a(2)=22. Presumably a(10)=196, as investigated
by Walker [2] and Irvin [3].
For further values, please see below. Can anyone confirm them?

I conjecture: a(n) is always positive, a(n)~n^2 and there are infinitely
many n, so that a(n)=n^2-n-1 (E.g. a(19)=19^2-19-1).
Furthermore, I bet there is always a set of sequences, which are transformed
under the 'reverse and add!'-process into each other. Just like the ones in
the proof by Brockhaus.

Best wishes and a happy new year

Frederick Magata
-

Below are the sequence details in the expected OEIS format:
%I A000001
%S A000001 22, 103, 290, 708, 1079, 2656, 1021, 593, 196, 1011, 237, 2701,
361, 447, 413, 3297, 519, 341, 379, 711, 461, 505, 551, 1022, 649, 701, 755,
811, 869, 929, 991, 1055, 1799, 1922, 1259, 1331, 1405, 1481, 1559, 1639,
1595, 1762, 1891, 1934, 2069, 2161, 2255, 2351, 2299, 2549, 4157, 2755,
2861, 2969, 3079, 3191, 3247, 3362, 3539, 3659, 3657, 3905, 4031, 4094,
3893, 4421, 4147, 4691, 4829, 7736, 8061, 8173, 5401, 5549, 5167, 5851,
5927, 6161, 6079, 6236, 6559, 6805, 6971, 6969, 7309, 6785, 7655, 7119,
8009, 8189, 8371, 8276, 8741, 8644, 8831, 8438, 8623, 9305, 9899
%N A000001 The minimal number a(n) so that the 'reverse and add!'-algorithm
in base n does not terminate in a palindrome. If there is no such number in
base n, then a(n):=-1.
%C A000001 All the values from this sequence, except the first one, are not
confirmed yet but only conjectured. (See [2] Walker, [3] Irvin on a(10)=196,
and [1] Brockhaus on a(2)=22)

An obvious algorithm is: Start with r:=n and check, wether the 'reverse and
add!'-algorithm in base n halts in a palindrome or not. If it stops,
increment r by one and repeat the process, else return r.
To obtain the values above, an upper limit of 100 'reverse and add!'-steps
was used, which seemed to suffice. I can not guarantee for it, though.

I conjecture: a(n) shows the same asymptotic behaviour as n^2. Additionally:
For infinite many n, we have a(n)=n^2-n-1.

Again, it is an open question, if the values of the sequence really lead to
infinitely many 'reverse and add!' steps or not. Furthermore: Is the
sequence always positive? I.e. has each base n a value a(n), so that the
'reverse and add!'-process never reaches a palindrome?
%e A000001 a(2)=22, see [1].
%Y A000001 Cf.
[1] K. Brockhaus, On the 'Reverse and Add!' algorithm in base 2
http://www.research.att.com/~njas/sequences/a058042.txt
[2] J. Walker, Three Years Of Computing: Final Report On The Palindrome
Quest
[3] T. Irvin, About Two Months of Computing, or An Addendum to Mr. Walker's
Three Years of Computing
%O A000001 2
%K A000001 ,look,nice,nonn,unkn,
%A A000001 Frederick Magata (frederick.magata at t-online.de), Dec 29 2001







From njas at research.att.com  Sun Dec 30 21:30:45 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Sun, 30 Dec 2001 15:30:45 -0500 (EST)
Subject: Math Gazette On-Line!
Message-ID: <200112302030.PAA97115@fry.research.att.com>


The wonderful English journal Math. Gazette has always been a great
source for sequences.  However, none of the local libraries here
subscribe to it.

Now Francisco Salinas (franciscodesalinas at hotmail.com) has discovered
that it is starting to appear on-line.  See

http://www.m-a.org.uk/eb/mg/index.htm

I encourage everyone to look at it - and to keep an
eye out for new sequences

NJAS




From njas at research.att.com  Mon Dec 31 19:26:08 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Mon, 31 Dec 2001 13:26:08 -0500 (EST)
Subject: making links to A-numbers
Message-ID: <200112311826.NAA32707@fry.research.att.com>

Thanks to Detlef Pauly for that new latex command
for making an A-number into a link.

I have two shell scripts that do similar things.
One makes A-numbers in a latex file into links,
the other does the same thing for html files.

Here they are (two separate files)

NJAS

--------------------------
# addlinks_tex.sh
# adds links to OEIS from A-numbers in a latex file,
# for use by the hyperef package
# revised Dec 9 2000

# check # of argts
if [ "$#" -eq 0 ]
then
echo "Incorrect no of argts. usage: addlinks_tex.sh file >out"
echo "Converts every sequence number A012345 etc into a link to the OEIS."
echo "Latex version"
exit 1
fi

cat $* | awk '
{ gsub( "A[0-9][0-9][0-9][0-9][0-9][0-9]",
"\\htmladdnormallink{&}{http:\/\/www.research.att.com\/cgi-bin\/access.cgi\/as\/njas\/sequences\/eisA.cgi?Anum=&}" )
print 
}

 '

----------------------------
# addlinks_html.sh
# adds links to OEIS from A-numbers in an html file
# revised Dec 9 2000

# check # of argts
if [ "$#" -eq 0 ]
then
echo "Incorrect no of argts. usage: addlinks_html.sh file >out"
echo "Converts every sequence number A012345 etc into a link to the OEIS."
echo "html version"
exit 1
fi

cat $* | awk '
{ gsub( "A[0-9][0-9][0-9][0-9][0-9][0-9]",
"<a href=\"http:\/\/www.research.att.com\/cgi-bin\/access.cgi\/as\/njas\/sequences\/eisA.cgi?Anum=&\">&<\/a>" )
print 
}
 '



MIME-Version: 1.0






From njas at research.att.com  Tue Dec 11 18:29:14 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Tue, 11 Dec 2001 12:29:14 -0500 (EST)
Subject: sum*product = n
Message-ID: <200112111729.MAA73530@fry.research.att.com>

Sol Golomb recently mentioned an old puzzle:

find all n such that sum of digits of n times
product of digits of n  is equal to n

this is a finite sequence which begins 1, 135, 144

does any seq fan know of further terms?

Neil

e g   (1+3+5)*1*3*5) = 135




From edp at wolfram.com  Tue Dec 11 18:57:20 2001
From: edp at wolfram.com (Ed Pegg Jr)
Date: Tue, 11 Dec 2001 11:57:20 -0600
Subject: Wilson's Corollary
In-Reply-To: <200112111729.MAA73530@fry.research.att.com>
Message-ID: <DCEIJGPGGGDPBLNBEBNBEEAACNAA.edp@wolfram.com>


  Last night, I looked at Wilson's Corollary.  I wrote
up the results at www.mathpuzzle.com .  There are many
sequences in this.  So far, I haven't found any in EIS,
but perhaps I'm looking in the wrong area.

  --Ed Pegg Jr.




From njas at research.att.com  Tue Dec 11 19:07:36 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Tue, 11 Dec 2001 13:07:36 -0500 (EST)
Subject: sum*product POSTSCRIPT
Message-ID: <200112111807.NAA26480@fry.research.att.com>

i should have looked harder 

this is A038369, which is labeled as "fini,full"

please ignore previous posting!




From rkg at cpsc.ucalgary.ca  Tue Dec 11 19:48:58 2001
From: rkg at cpsc.ucalgary.ca (Richard Guy)
Date: Tue, 11 Dec 2001 11:48:58 -0700 (MST)
Subject: sum*product POSTSCRIPT
In-Reply-To: <200112111807.NAA26480@fry.research.att.com>
Message-ID: <Pine.LNX.4.33.0112111138150.13879-100000@csl>

A small glitch in the following, which shd have 10^k - 1
instead of 10^(k-1).  This enables one to see that k < 60.
Still a bit big for a brute force search!
Note that if 5 occurs, then all digits are odd.
Also that, except possibly for one factor of size
< 9k, the number  n  is 7-smooth.       R.

%I A038369
%S A038369 0,1,135,144
%N A038369 n = (product of digits of n) * (sum of digits of n).
%C A038369 To prove finiteness, let k be the number of digits and consider 9^k*9*k < 10^(k-1) for every k > 100 - Ulrich Schimke (UlrSchimke at aol.com).
%H A038369 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Sum-ProductNumber.html">Link to a section of The World of Mathematics.</a>
%H A038369 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Digit.html">Link to a section of The World of Mathematics.</a>
%e A038369 144 belongs to the sequence because 1*4*4=16, 1+4+4=9 -> 16*9=144
%Y A038369 n = A007953(n) * A007954(n).
%K A038369 nice,nonn,fini,base,full,bref
%O A038369 1,3
%A A038369 Felice Russo (felice.russo at katamail.com)

On Tue, 11 Dec 2001, N. J. A. Sloane wrote:

> i should have looked harder 
> 
> this is A038369, which is labeled as "fini,full"





From njas at research.att.com  Tue Dec 11 20:10:20 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Tue, 11 Dec 2001 14:10:20 -0500 (EST)
Subject: sum*product POSTSCRIPT
Message-ID: <200112111910.OAA47069@fry.research.att.com>

Richard, I think 10^(k-1) was correct - the point being
that this is a LOWER bound on n.

anyway, i've replaced that comment with David Wilson's proof
that the sequence is complete





From rgwv at kspaint.com  Tue Dec 11 20:16:49 2001
From: rgwv at kspaint.com (Robert G. Wilson v)
Date: Tue, 11 Dec 2001 13:16:49 -0600
Subject: sum*product = n
References: <200112111729.MAA73530@fry.research.att.com>
Message-ID: <3C165BA1.EA55B4DA@kspaint.com>

Et al,

        Using Mathematica, I have tested the sequence up to 7.5*10^7 and
have found no further numbers which satisfy the criteria.

Bob.

"N. J. A. Sloane" wrote:

> Sol Golomb recently mentioned an old puzzle:
>
> find all n such that sum of digits of n times
> product of digits of n  is equal to n
>
> this is a finite sequence which begins 1, 135, 144
>
> does any seq fan know of further terms?
>
> Neil
>
> e g   (1+3+5)*1*3*5) = 135





From wilson at aprisma.com  Tue Dec 11 21:58:32 2001
From: wilson at aprisma.com (David W. Wilson)
Date: Tue, 11 Dec 2001 15:58:32 -0500
Subject: sum*product = n
References: <200112111729.MAA73530@fry.research.att.com>
    <3C165BA1.EA55B4DA@kspaint.com>
Message-ID: <3C167378.B424581@aprisma.com>

"Robert G. Wilson v" wrote:
> 
> Et al,
> 
>         Using Mathematica, I have tested the sequence up to 7.5*10^7 and
> have found no further numbers which satisfy the criteria.
> 
> Bob.
> 
> "N. J. A. Sloane" wrote:
> 
> > Sol Golomb recently mentioned an old puzzle:
> >
> > find all n such that sum of digits of n times
> > product of digits of n  is equal to n
> >
> > this is a finite sequence which begins 1, 135, 144
> >
> > does any seq fan know of further terms?
> >
> > Neil
> >
> > e g   (1+3+5)*1*3*5) = 135

I showed that 1, 135, and 144.  See "Sum-Product Number" at mathworld
for details.




From wilson at aprisma.com  Tue Dec 11 22:00:24 2001
From: wilson at aprisma.com (David W. Wilson)
Date: Tue, 11 Dec 2001 16:00:24 -0500
Subject: sum*product = n
References: <200112111729.MAA73530@fry.research.att.com>
    <3C165BA1.EA55B4DA@kspaint.com>
Message-ID: <3C1673E8.B1F54693@aprisma.com>

"Robert G. Wilson v" wrote:
> 
> Et al,
> 
>         Using Mathematica, I have tested the sequence up to 7.5*10^7 and
> have found no further numbers which satisfy the criteria.
> 
> Bob.
> 
> "N. J. A. Sloane" wrote:
> 
> > Sol Golomb recently mentioned an old puzzle:
> >
> > find all n such that sum of digits of n times
> > product of digits of n  is equal to n
> >
> > this is a finite sequence which begins 1, 135, 144
> >
> > does any seq fan know of further terms?
> >
> > Neil
> >
> > e g   (1+3+5)*1*3*5) = 135

Try again.

I showed that 1, 135, and 144 are all there are. 

See "Sum-Product Numbers" at mathworld.com.

- Dave Wilson
  Alzheimer's victim




From m.v.subbarao at ualberta.ca  Wed Dec 12 20:31:47 2001
From: m.v.subbarao at ualberta.ca (M.V Subbarao)
Date: Wed, 12 Dec 2001 12:31:47 -0700 (MST)
Subject: sumtimes product dividing n
Message-ID: <Pine.A41.4.33.0112121217190.31460-100000@gpu1.srv.ualberta.ca>


I wonder if there are infinitely many integers n such that the sum of its
digits times product of its digits is a divisor of n. On  example is n =
12. No doubt, we  can easily find many more.

M.V.Subbarao
m.v.subbarao at ualberta.ca





From rgwv at kspaint.com  Wed Dec 12 20:42:46 2001
From: rgwv at kspaint.com (Robert G. Wilson v)
Date: Wed, 12 Dec 2001 13:42:46 -0600
Subject: sumtimes product dividing n
References: <Pine.A41.4.33.0112121217190.31460-100000@gpu1.srv.ualberta.ca>
Message-ID: <3C17B335.52456749@kspaint.com>

Dear M.V.

        See
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=049102

Sincerely yours,
Robert G. Wilson, V

"M.V Subbarao" wrote:

> I wonder if there are infinitely many integers n such that the sum of its
> digits times product of its digits is a divisor of n. On  example is n =
> 12. No doubt, we  can easily find many more.
>
> M.V.Subbarao
> m.v.subbarao at ualberta.ca





From njas at research.att.com  Wed Dec 12 20:50:51 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Wed, 12 Dec 2001 14:50:51 -0500 (EST)
Subject: Jeroboam
Message-ID: <200112121950.OAA33220@fry.research.att.com>

A couple of weeks ago Marc LeBrun said

>  >=N. J. A. Sloane
>  > ...sequence number A065536...
> 
> Excellent!
> 
> In honor of this event, and in "thanksgiving" for your sterling
> proprietorship, on behalf of everyone I've taken the liberty of sending you
> a small token of all of our appreciation.  (Reassure your mail room; we're
> merely fanatics, not "sequence terrorists"!).
> 

I had forgotten all about this message but today the mail room
delivered one jeroboam of 21st Century Mumm Cuvee Napa.
Pretty amazing - never seen a jeroboam before
that wasn't locked up in a glass case.

Too bad the sequence fanatics are scattered over the 
whole world, otherwise we could have a party.  Email can't
do everything!

Thanks, Marc - and best wishes to everyone for 2002

Neil




From jawbrey at oakland.edu  Wed Dec 12 21:08:32 2001
From: jawbrey at oakland.edu (Jon Awbrey)
Date: Wed, 12 Dec 2001 15:08:32 -0500
Subject: Jeroboam
References: <200112121950.OAA33220@fry.research.att.com>
Message-ID: <3C17B940.88587978@oakland.edu>

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

|         ?
|        ?
|         .?
|        \_/
|         |
|        -^-
|
| e-bullient e-bubbly
|
| and cheers to all ...
|
| jon awbrey

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

N. J. A. Sloane wrote:
> 
> A couple of weeks ago Marc LeBrun said
> 
> > > N.J.A. Sloane
> > > ...sequence number A065536...
> >
> > Excellent!
> >
> > In honor of this event, and in "thanksgiving" for your sterling proprietorship,
> > on behalf of everyone I've taken the liberty of sending you a small token of
> > all of our appreciation.  (Reassure your mail room;  we're merely fanatics,
> > not "sequence terrorists"!).
> 
> I had forgotten all about this message but today the mail room
> delivered one jeroboam of 21st Century Mumm Cuvee Napa.
> Pretty amazing - never seen a jeroboam before
> that wasn't locked up in a glass case.
> 
> Too bad the sequence fanatics are scattered over
> the whole world, otherwise we could have a party.
> Email can't do everything!
> 
> Thanks, Marc - and best wishes to everyone for 2002
> 
> Neil

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?




From mlb at well.com  Wed Dec 12 21:49:45 2001
From: mlb at well.com (Marc LeBrun)
Date: Wed, 12 Dec 2001 12:49:45 -0800
Subject: Jeroboam
In-Reply-To: <200112121950.OAA33220@fry.research.att.com>
Message-ID: <5.1.0.14.2.20011212122511.00ab17b0@mail.well.com>

 >=N. J. A. Sloane
 > Too bad the sequence fanatics are scattered over the
 > whole world, otherwise we could have a party.

So we'll just have a distributed celebration.  Maybe convene a conference 
for A131072.  This will require a methuselah (although hopefully not a 
millennium!)

(More interesting numbers: "Methuselah was an antediluvian patriarch 
described in the Old Testament as having lived 969 years and whose name is 
synonymous with great age.  He may well have evolved from a character of 
earlier Sumerian legend who lived for 65,000 years.  To the Old Testament 
scribes this was perhaps too tall a tale, so they may have cut him back to 
a more conservative lifespan." --www.giantbottles.com)

Cheers!





From rpratt at email.unc.edu  Wed Dec 12 22:21:33 2001
From: rpratt at email.unc.edu (Rob Pratt)
Date: Wed, 12 Dec 2001 16:21:33 -0500 (EST)
Subject: antediluvian
In-Reply-To: <5.1.0.14.2.20011212122511.00ab17b0@mail.well.com>
Message-ID: <Pine.A41.4.21L1.0112121618390.50880-100000@login6.isis.unc.edu>

On Wed, 12 Dec 2001, Marc LeBrun wrote:

> ...
> 
> (More interesting numbers: "Methuselah was an antediluvian patriarch 
> described in the Old Testament as having lived 969 years and whose name is 
> synonymous with great age.  He may well have evolved from a character of 
> earlier Sumerian legend who lived for 65,000 years.  To the Old Testament 
> scribes this was perhaps too tall a tale, so they may have cut him back to 
> a more conservative lifespan." --www.giantbottles.com)
> 
> Cheers!

Note that Methuselah, Noah's grandfather, died in the year of The Flood
but wasn't on the ark.

Rob Pratt
Department of Operations Research
The University of North Carolina at Chapel Hill

rpratt at email.unc.edu

http://www.unc.edu/~rpratt/






From wilson at aprisma.com  Wed Dec 12 23:10:36 2001
From: wilson at aprisma.com (David W. Wilson)
Date: Wed, 12 Dec 2001 17:10:36 -0500
Subject: sumtimes product dividing n
References: <Pine.A41.4.33.0112121217190.31460-100000@gpu1.srv.ualberta.ca>
Message-ID: <3C17D5DC.EDE84D66@aprisma.com>

"M.V Subbarao" wrote:
> 
> I wonder if there are infinitely many integers n such that the sum of its
> digits times product of its digits is a divisor of n. On  example is n =
> 12. No doubt, we  can easily find many more.
> 
> M.V.Subbarao
> m.v.subbarao at ualberta.ca

Empirically, it looks as if every number of the form (10^3^n-1)/9 has the
desired property, which would make A049102 infinite.  Needs a proof, tho.




From mlb at well.com  Thu Dec 13 08:22:02 2001
From: mlb at well.com (Marc LeBrun)
Date: Wed, 12 Dec 2001 23:22:02 -0800
Subject: yet another unexpected EIS hit
Message-ID: <5.1.0.14.2.20011212223937.0408c178@mail.well.com>

Here's a pretty pair of dots that someone might be able to connect:

Define a "numbral arithmetic" by replacing addition with binary bitwise 
inclusive-OR (so that [3] + [5] = [7] etc) and multiplication becomes 
shift-&-OR instead of shift-&-add (so that [3] * [3] = [7] etc).

These numbrals have some interesting interpretations (such as a kind of 
"infinite base" system, or alternatively as sets of integers) that for 
brevity I'll resist  presenting.

Anyway, we can say naturally that [d] divides [n] when there exists an [e] 
such that

   [d] * [e] = [n].

For example the six divisors of [14] are [1], [2], [3], [6], [7] and [14].

Dot X: Counting the proper divisors of [2^n-1] gives the sequence

   0, 1, 2, 4, 7, 13, 23, 42, 76, 139, 255, 471, 873, 1627, 3044, 5718, ...

Dot Y: But this appears to be A048888, the anti-diagonal sums of table 
A048887 (qv)

            1  1  1  1  1  1  1 ...
            1  2  3  5  8 13 ...
            1  2  4  7 13 ...
            1  2  4  8 ...
            ...

where A(i,j) is the number of compositions of j into parts all <=i.

?!





From jawbrey at oakland.edu  Thu Dec 13 18:06:02 2001
From: jawbrey at oakland.edu (Jon Awbrey)
Date: Thu, 13 Dec 2001 12:06:02 -0500
Subject: lattice love song
Message-ID: <3C18DFFA.EF4AB088@oakland.edu>

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

seq fantasians,

did not know if folks had seen/heard this:

http://www.vub.ac.be/CLEA/aerts/latticelovesong.html

jon awbrey

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?




From njas at research.att.com  Sun Dec 16 06:39:02 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Sun, 16 Dec 2001 00:39:02 -0500 (EST)
Subject: report
Message-ID: <200112160539.AAA15347@fry.research.att.com>

There is a new version of the index, 
http://www.research.att.com/~njas/sequences/Sindx.html
which should be much faster

This involved many manual edits, so let me know if there are
errors or bad links

There will probably not be any further updates of the database
after tomorrow until the end of the year as I will be traveling
and unable to read email.  In case of emergency leave me voicemail
at (732) 828 6098.

Best wishes for the New Year!

Neil




From jawbrey at oakland.edu  Sun Dec 16 07:01:17 2001
From: jawbrey at oakland.edu (Jon Awbrey)
Date: Sun, 16 Dec 2001 01:01:17 -0500
Subject: report
References: <200112160539.AAA15347@fry.research.att.com>
Message-ID: <3C1C38AD.5DA2F17D@oakland.edu>

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

| Wollust ward dem Worm gegeben,
| Und der Cherub steht vor Gott!
|
| Friedrich von Schiller, Ode "An die Freude"

happy beethoven's birthday
and have a joyful new year!

jon awbrey

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

N. J. A. Sloane wrote:
> 
> There is a new version of the index,
> http://www.research.att.com/~njas/sequences/Sindx.html
> which should be much faster
> 
> This involved many manual edits, so let me know if there are
> errors or bad links
> 
> There will probably not be any further updates of the database
> after tomorrow until the end of the year as I will be traveling
> and unable to read email.  In case of emergency leave me voicemail
> at (732) 828 6098.
> 
> Best wishes for the New Year!
> 
> Neil

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?




From amarnath_murthy at yahoo.com  Sun Dec 16 08:18:23 2001
From: amarnath_murthy at yahoo.com (murthy amarnath)
Date: Sat, 15 Dec 2001 23:18:23 -0800 (PST)
Subject: report
In-Reply-To: <200112160539.AAA15347@fry.research.att.com>
Message-ID: <20011216071823.87356.qmail@web9604.mail.yahoo.com>


--- "N. J. A. Sloane" <njas at research.att.com> wrote:
> There is a new version of the index, 
>
http://www.research.att.com/~njas/sequences/Sindx.html
> which should be much faster
> 
> This involved many manual edits, so let me know if
> there are
> errors or bad links
> 
> There will probably not be any further updates of
> the database
> after tomorrow until the end of the year as I will
> be traveling
> and unable to read email.  In case of emergency
> leave me voicemail
> at (732) 828 6098.
> 
> Best wishes for the New Year!
> 
> Neil

Merry christmas and happy new year.
rgds
amarnath murthy


__________________________________________________
Do You Yahoo!?
Check out Yahoo! Shopping and Yahoo! Auctions for all of
your unique holiday gifts! Buy at http://shopping.yahoo.com
or bid at http://auctions.yahoo.com




From karttu at megabaud.fi  Mon Dec 17 19:00:32 2001
From: karttu at megabaud.fi (Antti Karttunen)
Date: Mon, 17 Dec 2001 20:00:32 +0200
Subject: An article in Frankfurter Allgemeine Zeitung
Message-ID: <3C1E32C0.E77F892E@megabaud.fi>


Jon Awbrey wrote:

  ?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

  | Wollust ward dem Worm gegeben,
  | Und der Cherub steht vor Gott!
  |
  | Friedrich von Schiller, Ode "An die Freude"

  happy beethoven's birthday
  and have a joyful new year!


Warm Christmas greetings from Finland also!


To delight your souls, I offer the article that appeared in 
Nr. 107 of Frankfurter Allgemeine Zeitung (09.05.2001),
as faithfully translated to English by
http://www.google.com/language_tools

To see the original Deutsch version, read:
http://www.megabaud.fi/~karttu/matikka/sloane.html

and this "English" version is also available as:
http://www.megabaud.fi/~karttu/matikka/sloane_eng.html

(I fetched the article from the electronic archives of "Frankfurter
Allgemeine"
http://afaz.gbi.de/ (and choose "Suche", then enter "Sloane" to "Suche"
field)
and it cost me just 1,50 euros, that is, a bit over one dollar).


Terveisin,

Antti Karttunen



   Nature and science Frankfurt general newspaper, 09,05,2001, NR. 107,
S. N1 
  
__________________________________________________________________________
                                                                         
   The passion of a number row collecting tank

   Data base with more than 61,000 entries/interest with mathematicians
and
   laymen largely 

   Some humans collect stamps, other coins, calling cards, beer covers
   or butterflies. There are hardly something, which did not become the
   object of human collecting passion, even night pots and tying lacings
   its lovers found. But for the probably most unusual collecting
objects
   the American mathematician Neil J. A has himself. Sloane of the AT&T
   Shannon lab in Florham park/new jersey decided. He collects
zahlenreihen.
   However not any arbitrary, but only such, which consist many elements
of
   positive whole numbers, are infinitely have and in addition according
to
   a firm rule developed.
  
   Although Sloane is probably the only collecting tank of zahlenreihen
in
   the world, its hobby encounters a broad interest. Thousands of
scientists
   and laymen help him for many years to constantly extend its
collection.
   In December 1963 Sloane, which was at this time still a student to
the
   Cornell University in Ithaca/New York, looked for information about a
   certain zahlenreihe from the graph theory. But as it also strove
itself,
   it could find nothing over it in the relevant literature. That
annoyed
   it so much that he began to collect systematically zahlenreihen.
 
   Later its collection over 2300 rows from all ranges of mathematics,
   the natural sciences and even the mental exercise covered ten years.
   It arranged it lexically and published it as book with the title
   "A Handbook OF Integer Sequences". The book became a success, and
many
   humans sent to it thereupon new rows. Neil Sloane continued to
collect.
   it wrote 1995 together with Simon Plouffe of the Universit? you
Qu?bec in
   Montr?al the "Encyclopedia OF Integer Sequences", which was than
twice
   as large with 5488 zahlenreihen more like its first collection.

   In the same year Sloane furnished E-Mail addresses, with which one
could
   make autopollings to its number row collection. The book and the
E-Mail
   addresses were a large success and led to an enormous tide of entries
   with new rows. One year later had already increased the collection on
   16,000 rows. Now Sloane arranged also its own InterNet side for its
   number row collection with special search functions
   ( http://www.research.att.com/~njas/sequences/ ).
   The interest among scientists and also among laymen is enormous. Per
day
   for instance 2500mal his collection one accesses, which contains in
the
   meantime over 61,000 rows.

   The collection of Sloane resembles a well sorted department store.
   All only somehow conceivable zahlenreihen are to be found there.
   Mathematical rows like those of the prime numbers (2, 3, 5, 7,
11...),
   the quadratzahlen (0, 1, 4, 9, 16...) or the faculties (1, 1, 2, 6,
24...)
   are naturally numerously represented. In addition, Neil Sloane seized
   numbers of chemistry like the number of the different alkanes with
   n carbon atoms (1, 1, 1, 2, 3, 5...) or numbers of physics like the
   number of the Feynman graphs of the order 2n (1, 3, 18, 153, 1638...)
   as well as numbers of biology like the possible secondary structures
   of a RNA molecule with n nucleotides (1, 1, 1, 2, 4, 8, 17...).

   In addition the collection contains chess problems like the number of
   the possibilities of placing n ladies in such a way on a chessboard
   with n fields that they do not threaten themselves mutually
   (1, 0, 0, 2, 10, 4, 40, 92...). To find curiosities are additional
   like the row 0, 0, 0, 0, 4, 9, 5, 1, 1, 0, 55... It results from the
   fact that one paints three, four, from the English number words one,
   two, five... all letters up to the number characters I, V, X, L, C, D
   and M. The word remainders are then interpreted as Roman numbers.

   In mystery columns and with intelligence tests question
   "as, liked, is called the next number?" a given zahlenreihe is easy
   with Sloanes collection to solve. If one enters the row 3, 1, 4, 1, 5
   for example into the search program, it offers to thirty-five
different
   possibilities, how the row could continue. One of the resulting rows
   would be from sequential whole numbers - beginning with the three -,
   which are separate by ones in each case. The next number would have
   to be thus unity. It could concern in addition, the decimal places of
the
   circle number of pi. Then the next element would have to be nine.
   Since 1998 the American mathematician even gives a special electronic
   magazine, which "journal OF Integer Sequences", out, in which
excluding
   articles over zahlenreihen appear.

   HEINRICH RESTRAIN

   All rights reserve. (C) F.A.Z . GmbH, Frankfurt/Main




From reinermartin at nyc.rr.com  Tue Dec 18 05:22:36 2001
From: reinermartin at nyc.rr.com (Reiner Martin)
Date: Mon, 17 Dec 2001 23:22:36 -0500
Subject: An article in Frankfurter Allgemeine Zeitung
References: <3C1E32C0.E77F892E@megabaud.fi>
Message-ID: <00e301c1877b$a2646460$6e7ba8c0@nyc.rr.com>


Since the automatic translation was a bit crude, I tried to translate this
article
myself as good as I could (my mother tongue is German). Here it is:

------------------------------------

Frankfurter Allgemeine Zeitung, May 9, 2001
Section: Nature and Science
Title: The Passion of a Integer Sequence Collector
Subtitle: Database with more than 61,000 entries / Wide interest with
mathematicians and amateurs


Some people collect stamps, other coins, calling cards, beer mats or
butterflies. There is hardly something which has not become the object of
the human passion to collect, even chamber pots and shoe laces have found
their devotees.  But the American mathematician Neil J. A. Sloane of the
AT&T Shannon Lab in Florham Park/New Jersey has probably chosen the most
unusual objects to collect.  He collects integer sequences.  Not any
arbitrary sequences however, but only such which consist of positive
integers, which have infinitely many elements, and which are build according
to a fixed rule.

Although Sloane is probably the only collector of integer sequences in the
world, his hobby is met with wide interest.  Thousands of scientists and
amateurs are helping him for many years now to continuously extend his
collection.  In December 1963 Sloane, who was at this time still a student
of Cornell University in Ithaca/New York, looked for information about a
certain sequence from graph theory.  But as hard as he tried, he could not
find anything about it in the relevant literature.  That annoyed him so much
that he began to collect sequences systematically.

Ten years later his collection contains over 2300 sequences from all areas
of mathematics, the natural sciences and even from puzzles.  He arranged
them lexically and published them as book with the title "A Handbook of
Integer Sequences."  The book became a success, and many people sent him new
sequences.  Neil Sloane continued to collect.  Together with Simon Plouffe
of the Universit? du Qu?bec in Montr?al he wrote in 1995 the "Encyclopedia
of Integer Sequences", which was with 5488 sequences more than twice as
large as his first collection.

In the same year Sloane created e-mail addresses with which one could make
automatic look-ups in his sequence database.  The book and the e-mail
addresses were a large success and led to an enormous wave of contributions
of new sequences.  One year later the collection had already increased to
16,000 sequences.  Then Sloane created also an Internet page for his
sequence database with special search functions
(www.research.att.com/~njas/sequences).  The interest among scientists and
also among amateurs is enormous.  Every day his collection is accessed about
2,500 times, which in the meantime contains over 61,000 sequences.

The collection of Sloane resembles a well sorted department store.  All
somehow conceivable sequences are to be found there.  Mathematical sequences
like those of the prime numbers (2, 3, 5, 7, 11 ...), the square numbers (0,
1, 4, 9, 16 ...) or the factorials (1, 1, 2, 6, 24 ...) are of course
represented numerously.  In addition, Neil Sloane added sequences from
chemistry like the number of different alkanes with n carbon atoms (1, 1, 1,
2, 3, 5 ...), or sequences from physics like the number of Feynman diagrams
of order 2n (1, 3, 18, 153, 1638...), as well as sequences from biology like
the possible secondary structures of a RNA molecule with n nucleotides (1,
1, 1, 2, 4, 8, 17 ...).

Additionally, the collection contains chess problems like the number of ways
of placing n queens on a chessboard with n by n squares in such a way that
they do not mutually attack themselves (1, 0, 0, 2, 10, 4, 40, 92 ...).  One
can also find curiosities like the sequence 0, 0, 0, 0, 4, 9, 5, 1, 1, 0, 55
...  It results from removing all letters except the number characters I, V,
X, L, C, D and M from the English numbers one, two, three, four, five, ...
The resulting words are then interpreted as Roman numbers.

The question "what is the next number?" of a given sequence, which is
popular in puzzle columns and intelligence tests, is easy to solve with
Sloanes collection.  For example, if one enters the sequence 3, 1, 4, 1, 5
into the search program, it offers thirty-five different possibilities to
continue the sequence.  One of the resulting sequences would be the one of
all integers, beginning with three, separated by ones.  Thus, the next
number would have to be one.  But it could also represent the decimals of
the number pi.  Then the next element would have to be nine.  Since 1998 the
American mathematician even publishes a special electronic magazine, the
"Journal of Integer Sequences", in contains exclusively articles on integer
sequences.

HEINRICH HEMME

Translated from the German by Reiner Martin

All rights reserved.  (C)  F.A.Z .  GmbH, Frankfurt/Main



----- Original Message -----
From: "Antti Karttunen" <karttu at megabaud.fi>
To: <seqfan at ext.jussieu.fr>
Sent: Monday, December 17, 2001 1:00 PM
Subject: An article in Frankfurter Allgemeine Zeitung


>
> Jon Awbrey wrote:
>
>   ?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?
>
>   | Wollust ward dem Worm gegeben,
>   | Und der Cherub steht vor Gott!
>   |
>   | Friedrich von Schiller, Ode "An die Freude"
>
>   happy beethoven's birthday
>   and have a joyful new year!
>
>
> Warm Christmas greetings from Finland also!
>
>
> To delight your souls, I offer the article that appeared in
> Nr. 107 of Frankfurter Allgemeine Zeitung (09.05.2001),
> as faithfully translated to English by
> http://www.google.com/language_tools
>
> To see the original Deutsch version, read:
> http://www.megabaud.fi/~karttu/matikka/sloane.html
>
> and this "English" version is also available as:
> http://www.megabaud.fi/~karttu/matikka/sloane_eng.html
>
> (I fetched the article from the electronic archives of "Frankfurter
> Allgemeine"
> http://afaz.gbi.de/ (and choose "Suche", then enter "Sloane" to "Suche"
> field)
> and it cost me just 1,50 euros, that is, a bit over one dollar).
>
>
> Terveisin,
>
> Antti Karttunen
>
>
>
>    Nature and science Frankfurt general newspaper, 09,05,2001, NR. 107,
> S. N1
>
> __________________________________________________________________________
>
>    The passion of a number row collecting tank
>
>    Data base with more than 61,000 entries/interest with mathematicians
> and
>    laymen largely
>
>    Some humans collect stamps, other coins, calling cards, beer covers
>    or butterflies. There are hardly something, which did not become the
>    object of human collecting passion, even night pots and tying lacings
>    its lovers found. But for the probably most unusual collecting
> objects
>    the American mathematician Neil J. A has himself. Sloane of the AT&T
>    Shannon lab in Florham park/new jersey decided. He collects
> zahlenreihen.
>    However not any arbitrary, but only such, which consist many elements
> of
>    positive whole numbers, are infinitely have and in addition according
> to
>    a firm rule developed.
>
>    Although Sloane is probably the only collecting tank of zahlenreihen
> in
>    the world, its hobby encounters a broad interest. Thousands of
> scientists
>    and laymen help him for many years to constantly extend its
> collection.
>    In December 1963 Sloane, which was at this time still a student to
> the
>    Cornell University in Ithaca/New York, looked for information about a
>    certain zahlenreihe from the graph theory. But as it also strove
> itself,
>    it could find nothing over it in the relevant literature. That
> annoyed
>    it so much that he began to collect systematically zahlenreihen.
>
>    Later its collection over 2300 rows from all ranges of mathematics,
>    the natural sciences and even the mental exercise covered ten years.
>    It arranged it lexically and published it as book with the title
>    "A Handbook OF Integer Sequences". The book became a success, and
> many
>    humans sent to it thereupon new rows. Neil Sloane continued to
> collect.
>    it wrote 1995 together with Simon Plouffe of the Universit? you
> Qu?bec in
>    Montr?al the "Encyclopedia OF Integer Sequences", which was than
> twice
>    as large with 5488 zahlenreihen more like its first collection.
>
>    In the same year Sloane furnished E-Mail addresses, with which one
> could
>    make autopollings to its number row collection. The book and the
> E-Mail
>    addresses were a large success and led to an enormous tide of entries
>    with new rows. One year later had already increased the collection on
>    16,000 rows. Now Sloane arranged also its own InterNet side for its
>    number row collection with special search functions
>    ( http://www.research.att.com/~njas/sequences/ ).
>    The interest among scientists and also among laymen is enormous. Per
> day
>    for instance 2500mal his collection one accesses, which contains in
> the
>    meantime over 61,000 rows.
>
>    The collection of Sloane resembles a well sorted department store.
>    All only somehow conceivable zahlenreihen are to be found there.
>    Mathematical rows like those of the prime numbers (2, 3, 5, 7,
> 11...),
>    the quadratzahlen (0, 1, 4, 9, 16...) or the faculties (1, 1, 2, 6,
> 24...)
>    are naturally numerously represented. In addition, Neil Sloane seized
>    numbers of chemistry like the number of the different alkanes with
>    n carbon atoms (1, 1, 1, 2, 3, 5...) or numbers of physics like the
>    number of the Feynman graphs of the order 2n (1, 3, 18, 153, 1638...)
>    as well as numbers of biology like the possible secondary structures
>    of a RNA molecule with n nucleotides (1, 1, 1, 2, 4, 8, 17...).
>
>    In addition the collection contains chess problems like the number of
>    the possibilities of placing n ladies in such a way on a chessboard
>    with n fields that they do not threaten themselves mutually
>    (1, 0, 0, 2, 10, 4, 40, 92...). To find curiosities are additional
>    like the row 0, 0, 0, 0, 4, 9, 5, 1, 1, 0, 55... It results from the
>    fact that one paints three, four, from the English number words one,
>    two, five... all letters up to the number characters I, V, X, L, C, D
>    and M. The word remainders are then interpreted as Roman numbers.
>
>    In mystery columns and with intelligence tests question
>    "as, liked, is called the next number?" a given zahlenreihe is easy
>    with Sloanes collection to solve. If one enters the row 3, 1, 4, 1, 5
>    for example into the search program, it offers to thirty-five
> different
>    possibilities, how the row could continue. One of the resulting rows
>    would be from sequential whole numbers - beginning with the three -,
>    which are separate by ones in each case. The next number would have
>    to be thus unity. It could concern in addition, the decimal places of
> the
>    circle number of pi. Then the next element would have to be nine.
>    Since 1998 the American mathematician even gives a special electronic
>    magazine, which "journal OF Integer Sequences", out, in which
> excluding
>    articles over zahlenreihen appear.
>
>    HEINRICH RESTRAIN
>
>    All rights reserve. (C) F.A.Z . GmbH, Frankfurt/Main






From karttu at megabaud.fi  Tue Dec 18 10:33:03 2001
From: karttu at megabaud.fi (Antti Karttunen)
Date: Tue, 18 Dec 2001 11:33:03 +0200
Subject: An article in Frankfurter Allgemeine Zeitung
References: <3C1E32C0.E77F892E@megabaud.fi>
    <00e301c1877b$a2646460$6e7ba8c0@nyc.rr.com>
Message-ID: <3C1F0D4F.C96EB42A@megabaud.fi>



Reiner Martin wrote:

> Since the automatic translation was a bit crude, I tried to translate this
> article
> myself as good as I could (my mother tongue is German).

Thanks!
I just tested the translation also with Altavista's Babelfish
at http://world.altavista.com/tr
and realized that its German-to-English translation almost
certainly uses the same software as Google at
http://www.google.com/language_tools
but with a slightly differing lexicon.


Terveisin,

Antti Karttunen






From njas at research.att.com  Tue Dec 18 15:11:05 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Tue, 18 Dec 2001 09:11:05 -0500 (EST)
Subject: FAZ
Message-ID: <200112181411.JAA65939@fry.research.att.com>

Thanks very much to  Antti Karttunen for the German version
and to Reiner Martin for the excellent translation of the F.A.Z. article.

I made a few small changes to the English version and put both versions
on the "Welcome to the OEIS" page (Seis.html) (with Reiner's permission)

NJAS




From layman at calvin.math.vt.edu  Tue Dec 18 19:11:31 2001
From: layman at calvin.math.vt.edu (John Layman)
Date: Tue, 18 Dec 2001 13:11:31 -0500 (EST)
Subject: More on "numbral" divisors
Message-ID: <20011218181145Z10637-16465+28@calvin.math.vt.edu>

On Dec 12, 2001, Marc LeBrun (mlb at mail.well.com) conjectured that 
the number of proper "numbral" divisors of 2^n-1 gives A048888, 
later confirmed by Richard Schroeppel (rcs at cs.arizona.edu).  
Calculations that I have made recently suggest the following possibly 
related conjecture concerning A007059 (balanced ordered trees with 
n nodes).

Conjecture: For n>=1, A007059(n+1) is the number of "numbral" divisors 
of (4^n-1)/3 = A002450(n).

Can anyone confirm this?

Further, I have shown that A002450(n) = (4^n-1)/3 is the nth umbral 
power of 5.





From karttu at megabaud.fi  Wed Dec 19 13:35:38 2001
From: karttu at megabaud.fi (Antti Karttunen)
Date: Wed, 19 Dec 2001 14:35:38 +0200
Subject: More on "numbral" divisors
References: <20011218181145Z10637-16465+28@calvin.math.vt.edu>
Message-ID: <3C20899A.5C80C901@megabaud.fi>



John Layman wrote:

> On Dec 12, 2001, Marc LeBrun (mlb at mail.well.com) conjectured that
> the number of proper "numbral" divisors of 2^n-1 gives A048888,
> later confirmed by Richard Schroeppel (rcs at cs.arizona.edu).

Dear John, Richard, others.

A little wish: Could you publish these little proofs somewhere
(e.g. as an associated notes file stored under EIS for the corresponding
sequence)?
I guess "HAKMEM" http://www.inwap.com/pdp10/hbaker/hakmem/hakmem.html
style would be fine.
Unless of course the proof is longer, and will be published in
a journal some day.

I think the value of EIS increases a year by year (contrary to
some other opinions...), especially when the important
sequences contain more and more references to the literature,
to the existing quality web sites (containing proofs also), and
to the other related EIS-sequences of the importance.



Merry christmas,

Antti Karttunen






From ogerard at ext.jussieu.fr  Wed Dec 19 15:34:45 2001
From: ogerard at ext.jussieu.fr (Olivier Gerard)
Date: Wed, 19 Dec 2001 15:34:45 +0100
Subject: Seqfan Site (was Re: More on "numbral" divisors)
In-Reply-To: <3C20899A.5C80C901@megabaud.fi>
References: <20011218181145Z10637-16465+28@calvin.math.vt.edu>
    <3C20899A.5C80C901@megabaud.fi>
Message-ID: <20011219143445.GD16003@ibazardev.ibazar-group.com>


To all seqfan members,

the site seqfan.net can be used by any EIS contributor to
store permanently any material related to sequences.

Material can be indexed by sequence number for urls

(ex http://www.seqfan.net/Axxxxxx/   )

or by a contributor log name.
As I can use symbolic links, the same material
can be attributed to several sequences.

Once more than one page is available I will make
two index pages available from the home page
for looking up material.

CGI/dynamic pages can be written in perl and php.

Just mail it to me for inclusion (please put Neil
in copy each time).

Olivier


On Wed, Dec 19, 2001 at 02:35:38PM +0200, Antti Karttunen wrote:
> 
> Dear John, Richard, others.
> 
> A little wish: Could you publish these little proofs somewhere
> (e.g. as an associated notes file stored under EIS for the corresponding
> sequence)?
> I guess "HAKMEM" http://www.inwap.com/pdp10/hbaker/hakmem/hakmem.html
> style would be fine.
> Unless of course the proof is longer, and will be published in
> a journal some day.
> 
> I think the value of EIS increases a year by year (contrary to
> some other opinions...), especially when the important
> sequences contain more and more references to the literature,
> to the existing quality web sites (containing proofs also), and
> to the other related EIS-sequences of the importance.
> 
> 
> 
> Merry christmas,
> 
> Antti Karttunen
> 
> 
> 




From karttu at megabaud.fi  Wed Dec 19 17:43:43 2001
From: karttu at megabaud.fi (Antti Karttunen)
Date: Wed, 19 Dec 2001 18:43:43 +0200
Subject: Seqfan Site
References: <20011218181145Z10637-16465+28@calvin.math.vt.edu>
    <3C20899A.5C80C901@megabaud.fi>
    <20011219143445.GD16003@ibazardev.ibazar-group.com>
Message-ID: <3C20C3BE.3D09F90B@megabaud.fi>


Dear Olivier,

seeing that there's not yet much activity in www.seqfan.net,
I suggest that you make the archives of SeqFan-list
(from  Listserv at Ext.jussieu.fr) available there
(either as a static Web-pages, or through some dynamic
script), because currently they are very difficult to access
via SeqFan-bitserv's own request-mechanism,
unless you want to delimit access only to SeqFan-members
themselves. Indeed, the publishing of e-mail addresses
occurring in the headers of SeqFan-messages
to any web-browsing robot might be a problem viz-a-viz spammers.
On the other hand, I guess that the most SeqFanatics
that have mailed to the list have already published their
mail-addresses in the entries they have submitted to EIS.

I guess the voice of the SeqFan-members should
be heard on this... Maybe one could use a "clever"
anti-spamming substitution like
sed -e 's/@/@supprimez./g' when transferring
the archives in bulk..
(This probably breaks some Mathematica code also,
so should preferably be applied only to the headers
and nearby of the messages.)

Salut,

  Antti


Olivier Gerard wrote:

> To all seqfan members,
>
> the site seqfan.net can be used by any EIS contributor to
> store permanently any material related to sequences.
>
> Material can be indexed by sequence number for urls
>
> (ex http://www.seqfan.net/Axxxxxx/   )
>
> or by a contributor log name.
> As I can use symbolic links, the same material
> can be attributed to several sequences.
>
> Once more than one page is available I will make
> two index pages available from the home page
> for looking up material.
>
> CGI/dynamic pages can be written in perl and php.
>
> Just mail it to me for inclusion (please put Neil
> in copy each time).
>
> Olivier
>





From frank.ellermann at t-online.de  Wed Dec 19 18:33:46 2001
From: frank.ellermann at t-online.de (Frank Ellermann)
Date: Wed, 19 Dec 2001 18:33:46 +0100
Subject: Seqfan Site
References: <20011218181145Z10637-16465+28@calvin.math.vt.edu>
    <3C20899A.5C80C901@megabaud.fi>
    <20011219143445.GD16003@ibazardev.ibazar-group.com>
    <3C20C3BE.3D09F90B@megabaud.fi>
Message-ID: <3C20CF7A.4DCA@t-online.de>

Hello Antti, you wrote:

> Indeed, the publishing of e-mail addresses
> occurring in the headers of SeqFan-messages
> to any web-browsing robot might be a problem
...
> I guess the voice of the SeqFan-members should
> be heard on this...

If the SeqFan archives are completely public, then
we could also organize it as newsgroup and retrieve
old stuff with Google.  Or in other words, I don't
like THIS idea, because I want to post really stupid
questions in SeqFan more privately, and not ready to 
be read by Google users in 2121... ;-)

But I like your original idea:  Collecting infos for
EIS sequences, especially sources used to calculate
terms in EIS sequences.  Probably you won't like my
REXX scripts like I can't use PARI, MAPLE, etc. here
directly, but often _any_ source is better than no
source at all.  Until now long programs are not shown
as %p in EIS, maybe seqfan.net files could solve this
problem ?

Example: Somebody computed A035615(26) this year, but
I don't know how.  All I have is my own algorithm and
some ideas, which needs about a day on my system to
verify A035615(16): yes, it's correct, but I want an
algorithm for more terms and not less... ;-)

                   Bye, Frank





From njas at research.att.com  Wed Dec 19 18:57:44 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Wed, 19 Dec 2001 12:57:44 -0500 (EST)
Subject: Seqfan Site
Message-ID: <200112191757.MAA34726@fry.research.att.com>


I am also perfectly willing to include plain text files 
on the sequence database web site, with links to them
from the apropriate entries.

There are already quite a few of these.  See for example

%H A005574 F. Ellermann, <a href="http://www.research.att.com/~njas/sequences/a002496.txt">Primes of the form (m^2)+1 up to 10^6</a>

%H A061396 V. Jovovic, <a href="http://www.research.att.com/~njas/sequences/a061396.txt">First 100 terms</a>

%H A006968 Gerard Schildberger, <a href="http://www.research.att.com/~njas/sequences/a006968.txt">The first 3999 numbers in Roman numerals</a>

%H A066057 K. Brockhaus, <a href="http://www.research.att.com/~njas/sequences/a058042.txt">On the'Reverse and Add!' algorithm in base 2</a>

etc

NJAS




From ogerard at ext.jussieu.fr  Wed Dec 19 20:26:28 2001
From: ogerard at ext.jussieu.fr (Olivier Gerard)
Date: Wed, 19 Dec 2001 20:26:28 +0100
Subject: Seqfan Site, Newsgroup, etc.
In-Reply-To: <3C20C3BE.3D09F90B@megabaud.fi>
References: <20011218181145Z10637-16465+28@calvin.math.vt.edu>
    <3C20899A.5C80C901@megabaud.fi>
    <20011219143445.GD16003@ibazardev.ibazar-group.com>
    <3C20C3BE.3D09F90B@megabaud.fi>
Message-ID: <20011219192628.GF16003@ibazardev.ibazar-group.com>


Dear Antti and all,

I have reserved and used as a front page the domain name seqfan.net
to have a place in common for us for *public* information about
seqfan but also sequences.

I welcome any idea on making this a better companion site to the EIS.
As I said before, I have large control over what can be put there,
and not only text files.

The mailing list is another thing altogether.  My already long
experience on newsgroups and mailing lists convinces me that
this list should stay private (i.e. no external posting allowed)
and as many of the members have subscribed knowing this was the
case, I have certainly no intent to post publicly any seqfan list
mail or email address without prior consent or initiative of the
members involved.

Now, the mail order archive system is cumbersome and without
visibility, especially for new members so I have prepared a private
browsable archive with monharc and a few other tools but requests
have been very seldom, and almost always from people able to
massage the archive in their own prefered format and tools.
So I have not put it online but I could if there is sufficient interest.
In this case I would make it restricted to seqfan members
so email harvesting would not be a concern.

We could consider making a newsgroup out of seqfan, but frankly
I think it would be a waste of time and peace.  Seqfan is already
very open (compare that to mathfun for the people who can), and
I never refuse someone who has made a contribution on the EIS.
My only requirement these days is to have a valid, non advertisement
postscripting email address and just to ask for it. This insures
the minimum level of motivation and interest to keep administrating
this list to a minimum compatable with my personal, academic and 
private life.

Like in a newsgroups, most members just read and don't contribute
but unlike most newsgroups, we don't need moderators and we don't 
experience flame wars.  The OEIS pages serve as a reference and
as a FAQ.  I am very much concerned about keeping the noise level
of seqfan as low as possible, and this not very easy in a newsgroup.

If you have reflexions on the newsgroup proposal or other modifications
of seqfan, please email them directly to Neil, me and Antti, because I don't
want to bother other members when possible on this meta-subject. 

Regards,

Olivier


On Wed, Dec 19, 2001 at 06:43:43PM +0200, Antti Karttunen wrote:
> 
> Dear Olivier,
> 
> seeing that there's not yet much activity in www.seqfan.net,
> I suggest that you make the archives of SeqFan-list
> (from  Listserv at Ext.jussieu.fr) available there
> (either as a static Web-pages, or through some dynamic
> script), because currently they are very difficult to access
> via SeqFan-bitserv's own request-mechanism,
> unless you want to delimit access only to SeqFan-members
> themselves. Indeed, the publishing of e-mail addresses
> occurring in the headers of SeqFan-messages
> to any web-browsing robot might be a problem viz-a-viz spammers.
> On the other hand, I guess that the most SeqFanatics
> that have mailed to the list have already published their
> mail-addresses in the entries they have submitted to EIS.
> 
> I guess the voice of the SeqFan-members should
> be heard on this... Maybe one could use a "clever"
> anti-spamming substitution like
> sed -e 's/@/@supprimez./g' when transferring
> the archives in bulk..
> (This probably breaks some Mathematica code also,
> so should preferably be applied only to the headers
> and nearby of the messages.)
> 
> Salut,
> 
>   Antti
> 




From mlb at well.com  Fri Dec 21 09:28:50 2001
From: mlb at well.com (Marc LeBrun)
Date: Fri, 21 Dec 2001 00:28:50 -0800
Subject: a tad more on OR-numbrals
Message-ID: <5.1.0.14.2.20011220231920.03c2ab48@mail.well.com>

A few tardy follow-on comments:

Many thanks to Rich Schroeppel for his clarifying explanation of the EIS 
hit.  I hope his "gapology" can be used to understand the full OR-numbral 
divisor sequence

   0 1 1 2 1 3 2 3 1 3 1 5 1 5 4 4 1 3 1 5 2 3 1 7 1 3 3 8 1 9 7 5 1 3 1 5 
1 3 1 7
   1 5 1 5 3 3 3 9 1 3 3 5 1 7 3 11 1 3 3 14 3 15 13 6 1 3 1 5 1 3 1 7 2 3 
1 5 1 3
   1 9 1 3 1 8 4 3 1 7 1 7 3 5 1 7 5 11 1 3 3 5 1 7 3 7 1 3 1 11 3 7 4 14 1 
3 3 5
   1 7 5 19 1 7 4...

which, along with a zillion others isn't in EIS yet because I can't fit the 
explanation in the "margin" (to address Antti Karttunen's complaint re. 
John Layman's interesting findings).  Maybe a numbral reference web page 
someday...

In the meantime, the two interpretations of OR-numbrals I alluded to are:

1. Sets of integers, with a 1 in the Nth bit denoting N's membership in the 
set.  Then numbral addition corresponds to set union, and multiplication 
means forming the set of all pair-wise sums.  (Maybe this would be useful 
for studying addition spectra or something).

2. Also, a homogeneous binary basis in powers of B, whose carries shift L 
places left, can be analyzed by solving
   B^N + B^N  =  B^(N+L)
to find the base B = 2^(1/L).  Thus the usual L=+1 gives vanilla binary 
B=2, L=-1 gives bit-reversed binary B=1/2, L=+2 gives the "tinker-toy" base 
B=sqrt(2), and so on.

When the carries don't shift at all addition degenerates into bitwise 
OR.  Here L=0, and we get B=2^(1/0).  So OR-numbrals are also a kind of 
"infinite" base.

But just what kind of "infinities" are these B^N=2^(N/0)?  Particularly 
that lsb "finity", 2^(0/0)?!

Can you unify these two interpretations?  A class of all sets of N things 
is somehow the same as some kind of Nth transfinite numeral?

(I suppose I should also mention that when you throw the carries away 
altogether you get XOR.  You might think of this as L=(minus) infinity, 
otherwise known as "polynomials over Z2" etc.  So perhaps either XOR is 
some kind of infinitesimal arithmetic, or else maybe a shaggy dog story 
about different flavors of zero...).

Each numbral system has its own "numbral theory" with analogs of 
partitions, divisors, etc that remain to be explored and sequenced.  If you 
come up with more such systems, interpretations and/or mysterious hits, 
please let me know.

Happy New Year!





From jens.voss at poet.de  Fri Dec 21 10:03:39 2001
From: jens.voss at poet.de (Jens Voss)
Date: Fri, 21 Dec 2001 10:03:39 +0100
Subject: a tad more on OR-numbrals
References: <5.1.0.14.2.20011220231920.03c2ab48@mail.well.com>
Message-ID: <3C22FAEB.29F85C96@poet.de>


Marc LeBrun wrote:
> 
> [...lots of interesting thoughts on different numbral systems]
> 

What strikes me most about the different numbral arithmetics is
that on one hand they are all nice since they are associative,
commutative and distributive, so the "divides" relation is
compatible with addition and multiplication. On the other hand,
the OR-numbral system appears to be the only one in which there
is no unique factorization, so as one consequence of that we have
to have to distinguish between "irreducible" and "prime" elements:

[5] for example is irreducible (since it is not a product of two
factors different from [1]), but not prime since it neither divides
[3] nor [11] but their product [3] * [11] = [31] = [5] * [7].

Gru?,
Jens

-- 
---------------------------------------------------------
Jens Voss, POET Software, Kattjahren 4 - 8, 22359 Hamburg
---------------------------------------------------------
The opinions expressed above are mine, not my employer's.
---------------------------------------------------------
"Tee-dah, tah-dee, tee-dah, tah-dee..."
                    J. Brahms, 4th symphony, 1st movement
---------------------------------------------------------




From njas at research.att.com  Sat Dec 22 19:09:38 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Sat, 22 Dec 2001 13:09:38 -0500 (EST)
Subject: small bug in %Y lines
Message-ID: <200112221809.NAA07269@fry.research.att.com>

There was a small bug in the program that processes new sequences and comments,
in the part that processes %Y or cross-reference lines.

The result was that in a few cases the whole %Y line was deleted.
-----------------------------------------------------------------

This only happened in the last two weeks, and did not
affect most submissions.

But if you kept a copy of recent submissions you might check to
see that the %Y lines were reproduced in the final version in the database.

The bug has now neen fixed.

I know it affected some recent submissions from Amarnath Murthy

Apologies

NJAS




From njas at research.att.com  Mon Dec 24 04:44:00 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Sun, 23 Dec 2001 22:44:00 -0500 (EST)
Subject: home for OEIS?
Message-ID: <200112240344.WAA82213@fry.research.att.com>

In view of recent developments at AT&T there is a possibility
that the On-Line Encyclopedia of Integer Sequences (and myself
- but that's another story) may need a new home one day.

One solution would be to get a domain name (OEIS.org, say) and
find an ISP to host it.  However, there are complications:
Size: over 256 MB
Operating system: all the lookup programs use Unix shell commands,
   superseeker also uses Maple, Mma, C, Fortran, etc
   so they would have to be available on the host machine.
Speed: needs a big fast Unix machine to get
   the rapid response we have now 

I have had no experience with ISP's.
If anyone has suggestions or advice please email me directly.
Neil Sloane, njas at research.att.com





From jawbrey at oakland.edu  Mon Dec 24 20:30:40 2001
From: jawbrey at oakland.edu (Jon Awbrey)
Date: Mon, 24 Dec 2001 14:30:40 -0500
Subject: Toward A Functional Conception Of Quantificational Logic
References: <3C21FA39.F189B7CC@oakland.edu>
    <3C220298.8D2147E4@oakland.edu> <3C251170.52243E10@oakland.edu>
    <3C255EAE.A4E5B20A@oakland.edu>
Message-ID: <3C278260.48418DBA@oakland.edu>

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

Note 128

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?

Subj:  Toward A Functional Conception Of Quantificational Logic

I am going to put off explaining Table 11, that presents a sample of
what I call "Interpretive Categories for Higher Order Propositions",
until after we get beyond the 1-dimensional case, since these lower
dimensional cases tend to be a bit "condensed" or "degenerate" in
their structures, and a lot of what is going on here will almost
automatically become clearer as soon as we get even two logical
variables into the mix.

| Document History:
|
| Subject:  Inquiry & Analogy
| Contact:  Jon Awbrey <jawbrey at oakland.edu>
| Version:  Draft 3.21
| Created:  01 Jan 1995
| Revised:  24 Dec 2001
| Faculty:  F. Mili & M.A. Zohdy
| Setting:  Oakland University, Rochester, Michigan, USA
| Excerpt:  Section 2.1.2 (Higher Order Propositions & Logical Operators)

2.1.2  Higher Order Propositions & Logical Operators (n = 2)

By way of reviewing notation and preparing to extend it to
higher order universes of discourse, let us first consider
the universe of discourse X? = [$X$] = [x_1, x_2] = [x, y],
based on two logical features or boolean variables x and y.

1.  The points of X? are collected in the space:

    X   =   <<x, y>>   =   {<x, y>}   ~=~   %B%^2.

    In other words, written out in full:

    X   =   {<"(x)", "(y)">,
             <"(x)", " y ">,
             <" x ", "(y)">,
             <" x ", " y ">}

    X  ~=~  {<%0%, %0%>,
             <%0%, %1%>,
             <%1%, %0%>,
             <%1%, %1%>}

2.  The propositions of X? make up the space:

    ^X^  =  (X -> %B%)  =  {f : X -> %B%}  ~=~  (%B%^2 -> %B%).

As always, it is frequently convenient to omit a few of the
finer markings of distinctions among isomorphic structures,
so long as one is aware of their presence and knows when
it is crucial to call upon them again.

The next higher order universe of discourse that is built on X? is
X?2 = [X?] = [[x, y]], which may be developed in the following way.
The propositions of X? become the points of X?2, and the mappings
of the type m : (X -> %B%) -> %B% become the propositions of X?2.
In addition, it is convenient to equip the discussion with with
a selected set of higher order operators on propositions, all
of which have the form w : (%B%^2 -> %B%)^k -> %B%.

To save a few words in the remainder of this discussion, I will
use the terms "measure" and "qualifier" to refer to all types of
"higher order" (HO) propositions and operators.  To describe the
present setting in picturesque terms, the propositions of [x, y]
may be regarded as a gallery of sixteen venn diagrams, while the
measures m : (X -> %B%) -> %B% are analogous to a body of judges
or a collection of critical viewers, each of whom evaluates each
picture as a whole and reports the ones that find favor or not.
In this way, each judge m_j partitions the gallery of pictures
into two aesthetic portions, the pictures (m_j)^(-1)(%1%) that
m_j likes and the pictures (m_j)^(-1)(%0%) that m_j dislikes.

There are 2^16 = 65536 measures of type m : (%B%^2 -> %B%) -> %B%.
Table 12 introduces the first 16 of these measures in the fashion
of the HO truth table that I used before.  The column headed "m_j"
shows the values of the measure m_j on each of the propositions
f_i : %B%^2 -> %B%, for i = 0 to 15, with blank entries in the
Table being optional for values of zero.  The arrangement of
measures that continues according to the plan indicated here
will be referred to as the "standard ordering" of measures.
In this scheme of things, the index j of the measure m_j is
the decimal equivalent of the bit string that is associated
with m_j's functional values, which can be obtained in turn
by reading the j^th column of binary digits in the Table as
the corresponding range of boolean values, taking them up
in the order from bottom to top.

Table 12.  Higher Order Propositions (n = 2)
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
|  | x | 1100 |    f     |m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|.|
|  | y | 1010 |          |0|0|0|0|0|0|0|0|0|0|1|1|1|1|1|1|.|
| f \  |      |          |0|1|2|3|4|5|6|7|8|9|0|1|2|3|4|5|.|
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
|      |      |          |                                 |
| f_0  | 0000 |    ()    |0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1  |
|      |      |          |                                 |
| f_1  | 0001 |  (x)(y)  |    1 1 0 0 1 1 0 0 1 1 0 0 1 1  |
|      |      |          |                                 |
| f_2  | 0010 |  (x) y   |        1 1 1 1 0 0 0 0 1 1 1 1  |
|      |      |          |                                 |
| f_3  | 0011 |  (x)     |                1 1 1 1 1 1 1 1  |
|      |      |          |                                 |
| f_4  | 0100 |   x (y)  |                                 |
|      |      |          |                                 |
| f_5  | 0101 |     (y)  |                                 |
|      |      |          |                                 |
| f_6  | 0110 |  (x, y)  |                                 |
|      |      |          |                                 |
| f_7  | 0111 |  (x  y)  |                                 |
|      |      |          |                                 |
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
|      |      |          |                                 |
| f_8  | 1000 |   x  y   |                                 |
|      |      |          |                                 |
| f_9  | 1001 | ((x, y)) |                                 |
|      |      |          |                                 |
| f_10 | 1010 |      y   |                                 |
|      |      |          |                                 |
| f_11 | 1011 |  (x (y)) |                                 |
|      |      |          |                                 |
| f_12 | 1100 |   x      |                                 |
|      |      |          |                                 |
| f_13 | 1101 | ((x) y)  |                                 |
|      |      |          |                                 |
| f_14 | 1110 | ((x)(y)) |                                 |
|      |      |          |                                 |
| f_15 | 1111 |   (())   |                                 |
|      |      |          |                                 |
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o

HO, HO, HO, ...

Jon Awbrey

?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?~~~~~~~~~?




From dp at dettodet.de  Thu Dec 27 11:09:14 2001
From: dp at dettodet.de (dp)
Date: Thu, 27 Dec 2001 11:09:14 +0100
Subject: A new LaTeX command: \anum{} and a comment
Message-ID: <MWMail.kfeacpce@host.none>



% ----------------------------------------------------------------
% LaTeX Paper *** anum.tex *** 2001-12-27
% Detlef Pauly, dp at dettodet.de
% **** -----------------------------------------------------------

\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage[colorlinks=true]{hyperref}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{8.9in}

% A new command: \anum{} -----------------------------------------

%with amsmath-package:
\newcommand{\anum}[1]
  {\text{\htmladdnormallink{A{#1}}
  {http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=#1}}}

%without amsmath-package:
%\newcommand{\anum}[1]
%  {\htmladdnormallink{A{#1}}
%  {http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=#1}}

% ----------------------------------------------------------------

\begin{document}

\textbf{A new \LaTeX\ command.}
\vspace{20pt}
\par\par
Hi seqfans.
\par
For example, to make a link to \anum{006125} use the small command

\begin{verbatim}
\anum{006125}
\end{verbatim}

instead of

\begin{footnotesize}
\begin{verbatim}
\htmladdnormallink{A006125}
{http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=006125}
\end{verbatim}
\end{footnotesize}

\par
Type 13 chars instead of 117 chars. Yep, $117 /13 = 9$.
\par

% ----------------------------------------------------------------

\vspace{30pt}
A new comment on \anum{006059} (Connected labeled topologies with $n$ points)
and \anum{001035} (Partially ordered sets [``posets''] with $n$ labeled elements
[or labeled acyclic transitive digraphs]):
%A006059(n)=n*A001035(n-1), n>0

\begin{equation*}
  \anum{006059}(n) = n \, \anum{001035}(n-1) \quad ,\text{for } n>0
\end{equation*}

\begin{small}
\anum{006059} 1, 1, 2, 9, 76, 1095, 25386, 910161, 49038872, 3885510411, \par
445110425110, 72721717736613, 16755380125270788, 5393244363726095487, 2405910197342218830914,
1477264863856923105482745, 1241074736327051013648799024, 1419169006353332682835352361843
\end{small}

% ----------------------------------------------------------------

\begin{table}[hb]
  \caption{Test}
  \label{tb:Test}
  \tabcolsep 4pt
  \renewcommand {\arraystretch} {1.3}
  \begin{small}
    \begin{tabular}{cccccccccc}
      \anum{000016} & \anum{000088} & \anum{000171} & \anum{000273} & \anum{000568} &
      \anum{000595} & \anum{000666} & \anum{000717} & \anum{000831} & \anum{001174} \\
      \anum{001187} & \anum{001349} & \anum{001437} & \anum{002499} & \anum{002500} &
      \anum{002785} & \anum{003027} & \anum{003030} & \anum{003085} & \anum{003086} \\
    \end{tabular}
  \end{small}
\end{table}

\par
Any comments/modifications?

\vspace{30pt}
ATB,
\par
DET \href{mailto:dp at dettodet.de}{dp at dettodet.de}

\end{document}

% ----------------------------------------------------------------





From Frederick.Magata at t-online.de  Sat Dec 29 07:17:09 2001
From: Frederick.Magata at t-online.de (Frederick Magata)
Date: Sat, 29 Dec 2001 07:17:09 +0100
Subject: 'reverse and add!'
Message-ID: <005101c19030$75764640$6e7ba8c0@ludwig>

Hello everyone,

this is my first contribution to the mailing list. So I hope you may find it
interesting.

In the context of the popular 'reverse and add!'-algorithm consider the
following sequence:
Let a(n) be the minimal number so that the 'reverse and add!'-algorithm in
base n does not terminate in a palindrome (in base n). If there is no such
number regarding base n, then a(n):=-1.

As proved by K. Brockhaus [1] a(2)=22. Presumably a(10)=196, as investigated
by Walker [2] and Irvin [3].
For further values, please see below. Can anyone confirm them?

I conjecture: a(n) is always positive, a(n)~n^2 and there are infinitely
many n, so that a(n)=n^2-n-1 (E.g. a(19)=19^2-19-1).
Furthermore, I bet there is always a set of sequences, which are transformed
under the 'reverse and add!'-process into each other. Just like the ones in
the proof by Brockhaus.

Best wishes and a happy new year

Frederick Magata
-

Below are the sequence details in the expected OEIS format:
%I A000001
%S A000001 22, 103, 290, 708, 1079, 2656, 1021, 593, 196, 1011, 237, 2701,
361, 447, 413, 3297, 519, 341, 379, 711, 461, 505, 551, 1022, 649, 701, 755,
811, 869, 929, 991, 1055, 1799, 1922, 1259, 1331, 1405, 1481, 1559, 1639,
1595, 1762, 1891, 1934, 2069, 2161, 2255, 2351, 2299, 2549, 4157, 2755,
2861, 2969, 3079, 3191, 3247, 3362, 3539, 3659, 3657, 3905, 4031, 4094,
3893, 4421, 4147, 4691, 4829, 7736, 8061, 8173, 5401, 5549, 5167, 5851,
5927, 6161, 6079, 6236, 6559, 6805, 6971, 6969, 7309, 6785, 7655, 7119,
8009, 8189, 8371, 8276, 8741, 8644, 8831, 8438, 8623, 9305, 9899
%N A000001 The minimal number a(n) so that the 'reverse and add!'-algorithm
in base n does not terminate in a palindrome. If there is no such number in
base n, then a(n):=-1.
%C A000001 All the values from this sequence, except the first one, are not
confirmed yet but only conjectured. (See [2] Walker, [3] Irvin on a(10)=196,
and [1] Brockhaus on a(2)=22)

An obvious algorithm is: Start with r:=n and check, wether the 'reverse and
add!'-algorithm in base n halts in a palindrome or not. If it stops,
increment r by one and repeat the process, else return r.
To obtain the values above, an upper limit of 100 'reverse and add!'-steps
was used, which seemed to suffice. I can not guarantee for it, though.

I conjecture: a(n) shows the same asymptotic behaviour as n^2. Additionally:
For infinite many n, we have a(n)=n^2-n-1.

Again, it is an open question, if the values of the sequence really lead to
infinitely many 'reverse and add!' steps or not. Furthermore: Is the
sequence always positive? I.e. has each base n a value a(n), so that the
'reverse and add!'-process never reaches a palindrome?
%e A000001 a(2)=22, see [1].
%Y A000001 Cf.
[1] K. Brockhaus, On the 'Reverse and Add!' algorithm in base 2
http://www.research.att.com/~njas/sequences/a058042.txt
[2] J. Walker, Three Years Of Computing: Final Report On The Palindrome
Quest
[3] T. Irvin, About Two Months of Computing, or An Addendum to Mr. Walker's
Three Years of Computing
%O A000001 2
%K A000001 ,look,nice,nonn,unkn,
%A A000001 Frederick Magata (frederick.magata at t-online.de), Dec 29 2001







From njas at research.att.com  Sun Dec 30 21:30:45 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Sun, 30 Dec 2001 15:30:45 -0500 (EST)
Subject: Math Gazette On-Line!
Message-ID: <200112302030.PAA97115@fry.research.att.com>


The wonderful English journal Math. Gazette has always been a great
source for sequences.  However, none of the local libraries here
subscribe to it.

Now Francisco Salinas (franciscodesalinas at hotmail.com) has discovered
that it is starting to appear on-line.  See

http://www.m-a.org.uk/eb/mg/index.htm

I encourage everyone to look at it - and to keep an
eye out for new sequences

NJAS




From njas at research.att.com  Mon Dec 31 19:26:08 2001
From: njas at research.att.com (N. J. A. Sloane)
Date: Mon, 31 Dec 2001 13:26:08 -0500 (EST)
Subject: making links to A-numbers
Message-ID: <200112311826.NAA32707@fry.research.att.com>

Thanks to Detlef Pauly for that new latex command
for making an A-number into a link.

I have two shell scripts that do similar things.
One makes A-numbers in a latex file into links,
the other does the same thing for html files.

Here they are (two separate files)

NJAS

--------------------------
# addlinks_tex.sh
# adds links to OEIS from A-numbers in a latex file,
# for use by the hyperef package
# revised Dec 9 2000

# check # of argts
if [ "$#" -eq 0 ]
then
echo "Incorrect no of argts. usage: addlinks_tex.sh file >out"
echo "Converts every sequence number A012345 etc into a link to the OEIS."
echo "Latex version"
exit 1
fi

cat $* | awk '
{ gsub( "A[0-9][0-9][0-9][0-9][0-9][0-9]",
"\\htmladdnormallink{&}{http:\/\/www.research.att.com\/cgi-bin\/access.cgi\/as\/njas\/sequences\/eisA.cgi?Anum=&}" )
print 
}

 '

----------------------------
# addlinks_html.sh
# adds links to OEIS from A-numbers in an html file
# revised Dec 9 2000

# check # of argts
if [ "$#" -eq 0 ]
then
echo "Incorrect no of argts. usage: addlinks_html.sh file >out"
echo "Converts every sequence number A012345 etc into a link to the OEIS."
echo "html version"
exit 1
fi

cat $* | awk '
{ gsub( "A[0-9][0-9][0-9][0-9][0-9][0-9]",
"<a href=\"http:\/\/www.research.att.com\/cgi-bin\/access.cgi\/as\/njas\/sequences\/eisA.cgi?Anum=&\">&<\/a>" )
print 
}
 '



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