[seqfan] A046988 query

[C Boyd]

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