[seqfan] Re: A046988 query

[C Boyd]

Dear Robert,

Thank you for explaining this.

If no one objects, I will draft a new sequence, "Numbers n that are not coprime to the numerator of zeta(2*n)/(Pi^(2*n))" in the next couple of days.

(I suppose A046988 has to be revised. The simplest approach would be to create a direct counterpart to A002432 by re-titling it as "Numerators of zeta(2*n)/Pi^(2*n)", removing A046988(0), and changing the offset to 1. In addition, a new sequence called "Numerators of Taylor series expansion of log(x/sin(x))" could be created. Editors please advise.)

I have calculated up to 10^5 those n that are not coprime to the numerator of zeta(2*n)/(Pi^(2*n)), and find they all continue to match A072823 (except for A072823(1)) as we have both suspected. It seems safe to conjecture that the sequence is identical to 2*A072823(n+1).

In addition, looking at the relevant GCDs, I conjecture that these are always powers of 2. For example (<= 10^5),

GCD = 2 for 14, 22, 26, 28, 30, 38, 42, 44, 46, 50, 52, 54, 56, 58, ...
GCD = 4 for 60, 92, 108, 116, 120, 124, 156, 172, 180, 184, 188, ...
GCD = 8 for 248, 376, 440, 472, 488, 496, 504, 632, 696, 728, 744, ...
GCD = 16 for 1008, 1520, 1776, 1904, 1968, 2000, 2016, 2032, 2544, ...
GCD = 32 for 4064, 6112, 7136, 7648, 7904, 8032, 8096, 8128, 8160

More conjectures. Taking GCDs vertically, (i) "14, 60, 248, 1008, 4064, ..." appears to be essentially the same as A171499 and A131262; (ii) "22, 92, 376, 1520, 6112, ..." appears to be essentially the same as A010036. Note that A131262 links back to sequences with a similar flavour to A046988. In A130654, Alexander Adamchuk conjectured that A092505(n) is always a power of 2, so there may be some underlying theme.

For those who are interested, here's a couple of Pari/GP snippets (adapting Lekraj Beedassy's formula for zeta(2*n)/(Pi^(2*n)) from A046988):

test(n)=if(gcd(numerator((-1)^(n+1)*2^(2*n-1)*bernfrac(2*n)/(2*n)!),n)!=1,1,0)
for(i=1,80,if(test(i),print1(i,", ")))
14, 22, 26, 28, 30, 38, 42, 44, 46, 50, 52, 54, 56, 58, 60, 62, 70, 74, 76, 78,

xgcd(n)=gcd(numerator((-1)^(n+1)*2^(2*n-1)*bernfrac(2*n)/(2*n)!),n)
for(i=1,90,g=xgcd(i);if(g!=1,print1(g,", ")))
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,

CB

-----Original Message-----
From: israel at math.ubc.ca  Mon Dec 21 22:10:02 2015
Date: 21 Dec 2015 14:09:54 -0800
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: A046988 query

Indeed. The coefficient of x^(2n) in log(x/sin(x)) is 
zeta(2*n)/(n*Pi^(2*n)). You'll run into trouble when n and the numerator of 
zeta(2*n)/Pi^(2*n) are not coprime, which is the case for n = 14, 22, 26, 
28, 30, 38, 42, 44, 46, 50, 52, 54, 56, 58, 60, 62, 70, 74, 76, 78, 82, 84, 
86, 88, 90, 92, 94, 98, 100, ... (a sequence which deserves its own OEIS 
entry, I think). Hmm: are these 2*A072823(k+1)?

Cheers,
Robert

On Dec 21 2015, C Boyd wrote:

>
>Dear SeqFans,
>
> A046988 is defined as "Numerators of Taylor series expansion of 
> log(x/sin(x)). For n>0, numerators of zeta(2*n)/Pi^(2*n)".
>
> Earlier I proposed an additional program and formula for A002432, and was 
> about to propose a similar program for A046988 when I noticed a potential 
> problem. The displayed terms and those in the b-file for A046988 match 
> (for n>0) the numerators of zeta(2*n)/Pi^(2*n): so far so good. However, 
> the first part of the definition does not always seem to generate those 
> terms. For example, using Pari/GP to calculate a(14) and a(22) according 
> to the first part,
>
>? default(seriesprecision, 99)
>
>? numerator(polcoeff(log(x/sin(x)),28))
>%1 = 3392780147
>
>? numerator(polcoeff(log(x/sin(x)),44))
>%2 = 2530297234481911294093
>
> The presented values for a(14) & a(22) are precisely twice these results, 
> namely 6785560294 & 5060594468963822588186.
>
> The indices <= 100 at which the sequences relating to the two definitions 
> differ appear to be: 14, 22, 26, 28, 30, 38, 42, 44, 46, 50, 52, 54, 56, 
> 58, 60, 62, 70, 74, 76, 78, 82, 84, 86, 88, 90, 92, 94, 98, 100. 
> Intriguingly, these seem to correspond exactly to 2*A072823(n) ("Numbers 
> that are not the sum of two powers of 2") for n>1, so if A046988 is 
> inconsistent, at least it is inconsistent in an interesting way.
>
>CB
>
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>
>

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