# "A dream" of a series :-)

Roland Bacher Roland.Bacher at ujf-grenoble.fr
Wed Jun 4 19:09:13 CEST 2008

```A somewhat similar product expansion is given by sequence A67856
yielding

exp(x)=product('(1+x^n)^(a(n)/n)','n'=1..infinity)

Roland Bacher

On Wed, Jun 04, 2008 at 03:56:35PM +0200, Gottfried Helms wrote:
> A dream of a series...
>
> Consider the exponentialseries
>
>    E0 =  1 + x/1! + x^2/2! + x^3/3!  + ...
>
> Because I considered the following problem in a roughly
> similar way,
>
>
> a factoring of the exponential-series.
>
> The idea due to owen james maresh ---------------------------------
>
> Define E1 by extracting the factor f1=(1+1*x):
>
>    E1 = E0 /(1+1*x) = 1 + 1/2*x^2 - 1/3*x^3 + 3/8*x^4 - 11/30*x^5 + 53/144*x^6
>
> Then define E2 by extracting the factor f2=(1+1/2*x^2):
>
>    E2 = 1 - 1/3*x^3 + 3/8*x^4 - 1/5*x^5 + 13/72*x^6
>
> Proceed this way to any arbitrary degree.
>
> -----
>
> Formally we get then (not discussing convergence)
>
>  E0 = f1 * f2 * f3 * f4 * ....
>     = (1+x)(1+1/2x^2)(1-1/3x^3) ...
>
> and the list of factors f1,f2,f3, begins then
>
>                                   x+1 = f1
>                             1/2*x^2+1 = f2
>                            -1/3*x^3+1 = f3
>                             3/8*x^4+1 = f4
>                            -1/5*x^5+1 = f5
>                           13/72*x^6+1
>                            -1/7*x^7+1
>                          27/128*x^8+1
>                           -8/81*x^9+1
>                         91/800*x^10+1
>                          -1/11*x^11+1
>                     1213/13824*x^12+1
>                          -1/13*x^13+1
>                       505/6272*x^14+1
>                    -1919/30375*x^15+1
>                     2955/32768*x^16+1
>                          -1/17*x^17+1
>                   24557/419904*x^18+1
>                          -1/19*x^19+1
>               1136313/20480000*x^20+1
>                  -34943/750141*x^21+1
>                   12277/247808*x^22+1
>                          -1/23*x^23+1
>            65978519/1528823808*x^24+1
>                     -624/15625*x^25+1
>                  57331/1384448*x^26+1
>                 -58528/1594323*x^27+1
>           195948483/5035261952*x^28+1
>                          -1/29*x^29+1
>   1052424027703/30233088000000*x^30+1
>
> The observation, which this correspondent stumbled across, was,
> that apparently at all prime indexes k the cofactor c_k has
> prime denominator, which alone is already interesting.
>
> -------------------------------------------------------------------
>
> But there's some more in it, as I just found with a short
> analysis (looking at it up to index n=64)
>
> Consider indexes composite of primes to the first power:
>    index       denominator
>   6 = 2*3       2^3 * 3^2
>  10 = 2*5       2^5 * 5^2
>  14 = 2*7       2^7 * 7^2
>
> index          denominator
>  30 = 2*3*5     2^(3*5) * 3^(2*5) * 5^(2*3)
>
> So we may rewrite
> index          denominator
> prime p = 1*p   p^1 * 1^p
>
>
> For prime-powers it looks like
>   4  = 2^2      2^3
>   8  = 2^3      2^7
>  16  = 2^4      2^15
>  32  = 2^5      2^31
>  64  = 2^6      2^63
>
>   9  = 3^2      3^4
>  27  = 3^3      3^13
>
>  25  = 5^2      5^6
>
>  49  = 7^2      7^8
>
> and some more composites - but here I don't really have an idea:
>
>  12  = 2^2*3    (2^3)^3 * 3^3
>  24  = 2^3*3    (2^7)^3 *(3^3)^2
>  48  = 2^4*3    (2^15)^3* (3^3)^5
>
>  18  = 2 * 3^2  2^6     * (3^4)^2
>  54  = 2 * 3^3  2^23    * (3^13)^2
>
>  36  = 2^2*3^2  2^24    * 3^15
>
>  25  = 5^2              5^6
>  50  = 2*5^2    2^21 * (5^6)^2
>
>  45  = 3^2 * 5  (3^4)^5 * 5^9
>  63  = 3^2 * 7  (3^4)^7 * 7^9
>
> Although I could not decode the composites reasonably, I feel
> this series of denominators is really "a dream" for a
> true seqfan... :-)
>
> Gottfried Helms

Dear Seqfans, 2 things about the OEIS Welcome Page:

1. A year or two ago someone designed a "search bar" plugin for the OEIS.
I could never get it to work.  Just now I tried again, and it works.
- I hope what I say is correct!
- I would like to give credit for this - who wrote the "oeis.xml" plugin?

2. On Mar 31 there was a discussion (see below) about subscribing
to an OEIS entry, so you are told if it changes.  I would like
to mention this in the Welcome page too.  But I cannot get it to work.

Could someone write me a paragraph about how to do this,
that I could (a) use myself and (b) add to the Welcome Page?

What I tried was this: I went to a sequence, say A000010.
Then I clicked Bookmarks (in Firefox), but the "Subscribe to This Page"
line was not activated (it was shadowy)

Thanks!

Neil

> Date: Mon, 31 Mar 2008 10:02:57 -0500
> Subject: "Subscribing" to an OEIS page
> From: Jonathan Sondow <jsondow at alumni.princeton.edu>
> To: SeqFans <seqfan at ext.jussieu.fr>
>
> Dear SeqFans,
>
> With Internet Explorer 5.1 for Mac, you can "subscribe" to any Web page.
> When you subscribe to a page, IE monitors the page and notifies you when the
> page is updated.
>
> Subscribing to OEIS pages should be a useful way to monitor them for
> changes. However, when I subscribe to the page of a sequence, IE notifies me
> every day that it has changed, even when it has (apparently) not changed.
>
> Can anyone explain this and/or suggest a fix?
>
> Thanks,
> Jonathan
>
> Date: Mon, 31 Mar 2008 10:36:14 -0400
> From: "Maximilian Hasler" <maximilian.hasler at gmail.com>
> To: "Jonathan Sondow" <jsondow at alumni.princeton.edu>
> Subject: Re: "Subscribing" to an OEIS page
>
> 1/ in Firefox this feature is of course also implemented (2nd link in
> Bookmarks menu in my version 2.0.0.13)
>
> 2/ in fact your IE does not know which part of the page is relevant
> data. On the bottom of each page is written how many sequences are in
> OEIS and when it has been updated; also the time needed to create the
> page.
>
> One way to fix this would be to subscribe to the text format, e.g.
> http://www.research.att.com/~njas/sequences/?q=id%3aA137999&p=1&n=10&fmt=3
> this also includes the time needed for the search, but with some luck,
> it will always be
> "Search completed in 0.001 seconds"
> if you just request one "id:xxx"
>
> Maximilian
>
> Date: Mon, 31 Mar 2008 16:34:10 +0200
> From: Richard Mathar <mathar at strw.leidenuniv.nl>
> Message-Id: <200803311434.m2VEYASM003206 at amer.strw.leidenuniv.nl>
> To: seqfan at ext.jussieu.fr
> Subject: Re: "Subscribing" to an OEIS page
>
> If this subscription is to an URL of the format
> http://research.att.com/~njas/sequences/Axxxxxx
> then this web page contains a bottom line similar to
> If IE compares web pages on a simple textual basis, the web page indeed
> changes about once a day when it is re-written some-when at 3 a.m. EDT.
> In that sense, IE is expected to say that it changed.
>
> RJM
>

Neil,

Re subscribing to an OEIS entry:

I followed Maximilian's suggestion and used IE to subscribe to the text
format of an OEIS entry. But I was often notified that the page had changed
when only the time to create the page had changed. So I stopped subscribing.

Best,
Jonathan

On 6/4/08 1:39 PM N. J. A. Sloane wrote:

>
> Dear Seqfans, 2 things about the OEIS Welcome Page:
>
> 1. A year or two ago someone designed a "search bar" plugin for the OEIS.
> I could never get it to work.  Just now I tried again, and it works.
> - I hope what I say is correct!
> - I would like to give credit for this - who wrote the "oeis.xml" plugin?
>
> 2. On Mar 31 there was a discussion (see below) about subscribing
> to an OEIS entry, so you are told if it changes.  I would like
> to mention this in the Welcome page too.  But I cannot get it to work.
>
> Could someone write me a paragraph about how to do this,
> that I could (a) use myself and (b) add to the Welcome Page?
>
> What I tried was this: I went to a sequence, say A000010.
> Then I clicked Bookmarks (in Firefox), but the "Subscribe to This Page"
> line was not activated (it was shadowy)
>
> Thanks!
>
> Neil
>
>
>> Date: Mon, 31 Mar 2008 10:02:57 -0500
>> Subject: "Subscribing" to an OEIS page
>> From: Jonathan Sondow <jsondow at alumni.princeton.edu>
>> To: SeqFans <seqfan at ext.jussieu.fr>
>>
>> Dear SeqFans,
>>
>> With Internet Explorer 5.1 for Mac, you can "subscribe" to any Web page.
>> When you subscribe to a page, IE monitors the page and notifies you when the
>> page is updated.
>>
>> Subscribing to OEIS pages should be a useful way to monitor them for
>> changes. However, when I subscribe to the page of a sequence, IE notifies me
>> every day that it has changed, even when it has (apparently) not changed.
>>
>> Can anyone explain this and/or suggest a fix?
>>
>> Thanks,
>> Jonathan
>>
>> Date: Mon, 31 Mar 2008 10:36:14 -0400
>> From: "Maximilian Hasler" <maximilian.hasler at gmail.com>
>> To: "Jonathan Sondow" <jsondow at alumni.princeton.edu>
>> Subject: Re: "Subscribing" to an OEIS page
>>
>> 1/ in Firefox this feature is of course also implemented (2nd link in
>> Bookmarks menu in my version 2.0.0.13)
>>
>> 2/ in fact your IE does not know which part of the page is relevant
>> data. On the bottom of each page is written how many sequences are in
>> OEIS and when it has been updated; also the time needed to create the
>> page.
>>
>> One way to fix this would be to subscribe to the text format, e.g.
>> http://www.research.att.com/~njas/sequences/?q=id%3aA137999&p=1&n=10&fmt=3
>> this also includes the time needed for the search, but with some luck,
>> it will always be
>> "Search completed in 0.001 seconds"
>> if you just request one "id:xxx"
>>
>> Maximilian
>>
>> Date: Mon, 31 Mar 2008 16:34:10 +0200
>> From: Richard Mathar <mathar at strw.leidenuniv.nl>
>> Message-Id: <200803311434.m2VEYASM003206 at amer.strw.leidenuniv.nl>
>> To: seqfan at ext.jussieu.fr
>> Subject: Re: "Subscribing" to an OEIS page
>>
>> If this subscription is to an URL of the format
>> http://research.att.com/~njas/sequences/Axxxxxx
>> then this web page contains a bottom line similar to
>> If IE compares web pages on a simple textual basis, the web page indeed
>> changes about once a day when it is re-written some-when at 3 a.m. EDT.
>> In that sense, IE is expected to say that it changed.
>>
>> RJM
>>
>

```