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%I #32 Aug 21 2023 11:55:41
%S 1,63,45991,2942695,45982595359,5109066151,601081707598999,
%T 38469080386820311,252396118308232060471,252395862211967012407,
%U 447134922152359540530757327,447134770212444455649757327,2158234586764514215343657417779543,308319185132349039219686748825649
%N Numerator of Sum_{k=1..n} (-1)^(k+1)/k^6.
%C p divides a(p-1) for prime p > 2. a(n) is prime for n = {19, 47, 164, ...} = A136686.
%C Lim_{n -> infinity} a(n)/A334605(n) = A275703 = (31/32)*A013664. - _Petros Hadjicostas_, May 07 2020
%H Harvey P. Dale, <a href="/A136677/b136677.txt">Table of n, a(n) for n = 1..382</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%e The first few fractions are 1, 63/64, 45991/46656, 2942695/2985984, 45982595359/46656000000, 5109066151/5184000000, ... = a(n)/A334605(n). - _Petros Hadjicostas_, May 07 2020
%t Table[ Numerator[ Sum[ (-1)^(k+1)/k^6, {k,1,n} ] ], {n,1,30} ]
%t Accumulate[Table[(-1)^(k+1)/k^6,{k,20}]]//Numerator (* _Harvey P. Dale_, Aug 21 2023 *)
%Y Cf. A013664, A058313, A119682, A120296, A136675, A136676.
%Y Cf. A001008, A007406, A007408, A007410, A099828, A103345, A275703.
%Y Cf. A136681, A136682, A136683, A136684, A136685, A136686, A334605 (denominators).
%K frac,nonn
%O 1,2
%A _Alexander Adamchuk_, Jan 16 2008