[seqfan] Re: A046988 query
israel at math.ubc.ca
israel at math.ubc.ca
Mon Dec 21 23:09:54 CET 2015
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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