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%I #16 Aug 11 2024 18:47:54
%S 1,257,1686433,431733409,168646292872321,168646392872321,
%T 972213062238348973121,248886558707571775009601,
%U 1632944749460578249437992161,1632944765723715465050248417
%N Numerators of sum_{k=1..n} 1/k^8 = Zeta(8,n).
%C a(n) gives the partial sums, Zeta(8,n) of Euler's Zeta(8). Zeta(k,n) is also called H(k,n) because for k=1 these are the harmonic numbers H(n) A001008/A002805.
%C For the denominators see A103350 and for the rationals Zeta(8,n) see the W. Lang link under A103345.
%F a(n)=numerator(sum_{k=1..n} 1/k^8).
%F G.f. for rationals Zeta(8, n): polylogarithm(8, x)/(1-x).
%t s=0;lst={};Do[s+=n^1/n^9;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jan 24 2009 *)
%t Table[ HarmonicNumber[n, 8] // Numerator, {n, 1, 10}] (* _Jean-François Alcover_, Dec 04 2013 *)
%t Accumulate[1/Range[10]^8]//Numerator (* _Harvey P. Dale_, Aug 11 2024 *)
%Y For k=1..7 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346, A103347/A103348.
%K nonn,frac,easy
%O 1,2
%A _Wolfdieter Lang_, Feb 15 2005