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%I #33 May 10 2020 19:56:43
%S 1,65,47449,3037465,47463376609,47464376609,5584183099672241,
%T 357389058474664049,260537105518334091721,52107472322919827957,
%U 92311616995117182948130877,92311647383100199924330877,445570781131605573859221176881493,445570839299219762020391212081493
%N Numerator of Sum_{k=1..n} 1/k^6 = Zeta(6,n).
%C For the rationals Zeta(k,n) for k = 1..10 and n = 1..20, see the W. Lang link.
%C a(n) gives the partial sum, Zeta(6,n), of Euler's (later Riemann's) Zeta(6). Zeta(k,n), k >= 2, is sometimes also called H(k,n) because for k = 1 these would be the harmonic numbers A001008/A002805. However, H(1,n) does not give partial sums of a convergent series.
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Wolfdieter Lang, <a href="/A103345/a103345.pdf">Rational Zeta(k,n) and more</a>.
%F a(n) = numerator(Sum_{k=1..n} 1/k^6) = numerator(A291456(n)/(n!)^6).
%F G.f. for rationals Zeta(6, n): polylogarithm(6, x)/(1-x).
%F Zeta(6, n) = (1/945)*Pi^6 - psi(5, n+1)/5!, see eq. 6.4.3 with m = 5, p. 260, of the Abramowitz-Stegun reference. - _Wolfdieter Lang_, Dec 03 2013
%e The first few fractions are 1, 65/64, 47449/46656, 3037465/2985984, 47463376609/46656000000, ... = A103345/A103346. - _Petros Hadjicostas_, May 10 2020
%t s=0; lst={}; Do[s+=n^1/n^7; AppendTo[lst,Numerator[s]],{n,3*4!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Jan 24 2009 *)
%t Table[ HarmonicNumber[n, 6] // Numerator, {n, 1, 12}] (* _Jean-François Alcover_, Dec 04 2013 *)
%Y Cf. A013664, A291456. For the denominators, see A103346.
%Y For k=1..5, see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052.
%K nonn,frac,easy
%O 1,2
%A _Wolfdieter Lang_, Feb 15 2005
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