%I #38 Jun 07 2021 04:46:21

%S 1,31,7565,241837,755989457,755889457,12705011703799,406547611705943,

%T 98792790681344149,98791774426324117,15910615688635928566967,

%U 15910549913780913466967,5907492176026179821253778331

%N Numerator of Sum_{k=1..n} (-1)^(k+1)/k^5.

%C a(n) is prime for n in A136685.

%C Lim_{n -> infinity} a(n)/A334604(n) = A267316 = (15/16)*A013663. - _Petros Hadjicostas_, May 07 2020

%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, 31/32, 7565/7776, 241837/248832, 755989457/777600000, 755889457/777600000, ... = a(n)/A334604(n). - _Petros Hadjicostas_, May 07 2020

%t Table[ Numerator[ Sum[ (-1)^(k+1)/k^5, {k,1,n} ] ], {n,1,30} ]

%o (PARI) a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^5)); \\ _Michel Marcus_, May 07 2020

%Y Cf. A001008, A007406, A007408, A007410, A013663, A058313, A099828, A103345, A119682, A120296, A136675, A136677, A136681, A136682, A136683, A136684, A136685, A136686, A267316, A334604 (denominators).

%K frac,nonn

%O 1,2

%A _Alexander Adamchuk_, Jan 16 2008