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%I #49 Nov 13 2023 09:07:42
%S 1,7,197,1549,195353,194353,66879079,533875007,14436577189,
%T 14420574181,19209787242911,19197460851911,42198121495296467,
%U 6025866788581781,6027847576222613,48209723660000029,236907853607882606477
%N Numerator of Sum_{k=1..n} (-1)^(k+1)/k^3.
%C a(n) is prime for n in A136683.
%C Lim_{n -> infinity} a(n)/A334582(n) = A197070. - _Petros Hadjicostas_, May 07 2020
%H Robert Israel, <a href="/A136675/b136675.txt">Table of n, a(n) for n = 1..768</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_eta_function">Dirichlet eta function</a>.
%e The first few fractions are 1, 7/8, 197/216, 1549/1728, 195353/216000, 194353/216000, 66879079/74088000, 533875007/592704000, ... = a(n)/A334582(n). - _Petros Hadjicostas_, May 06 2020
%p map(numer,ListTools:-PartialSums([seq((-1)^(k+1)/k^3, k=1..100)])); # _Robert Israel_, Nov 09 2023
%t (* Program #1 *) Table[Numerator[Sum[(-1)^(k+1)/k^3, {k,1,n}]], {n,1,50}]
%t (* Program #2 *) Numerator[Accumulate[Table[(-1)^(k+1) 1/k^3, {k,50}]]] (* _Harvey P. Dale_, Feb 12 2013 *)
%o (PARI) a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^3)); \\ _Michel Marcus_, May 07 2020
%Y Cf. A001008, A007406, A007408, A007410, A058313, A099828, A103345, A119682, A120296, A136676, A136677, A136681, A136682, A136683, A136684, A136685, A136686, A197070, A334582 (denominators).
%K frac,nonn
%O 1,2
%A _Alexander Adamchuk_, Jan 16 2008
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