"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,
>
> see http://groups.google.com/group/alt.math.recreational/browse_thread/thread/606d98fef53db8ff/7aa1d8f1f840736e?lnk=st&q=Gottfried+Helms+infinite+product+exp#7aa1d8f1f840736e
>
> a correspondent asked me, whether I knew the following about
> 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.
So I've added a comment about this to the Welcome Page.
- 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
> "..Last modified March 31 ... EDT ..... Contains xxxxx sequences".
> 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.
> So I've added a comment about this to the Welcome Page.
> - 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
>> "..Last modified March 31 ... EDT ..... Contains xxxxx sequences".
>> 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
>>
>
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