%I A103345
%S A103345 1,65,47449,3037465,47463376609,47464376609,5584183099672241,
%T A103345 357389058474664049,260537105518334091721,52107472322919827957,
%U A103345 92311616995117182948130877,92311647383100199924330877
%N A103345 Numerators of sum(1/k^6,k=1..n)=:Zeta(6,n).
%C A103345 For the denominators see A103346. For the rationals Zeta(k,n) for k=1..10, 
               n=1..20, see the W. Lang link.
%C A103345 a(n) gives the partial sums, 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 A103345 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A103345.text">
               Rational Zeta(k,n).</a>
%F A103345 a(n)=numerator(sum(1/k^6, k=1..n)).
%F A103345 G.f. for rationals Zeta(6, n): polylogarithm(6, x)/(1-x).
%t A103345 s=0;lst={};Do[s+=n^1/n^7;AppendTo[lst,Numerator[s]],{n,3*4!}];lst [From 
               Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 24 2009]
%Y A103345 For k=1..5 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/
               A007480, A099828/A069052.
%Y A103345 Sequence in context: A093265 A120801 A084225 this_sequence A061688 A015072 
               A015039
%Y A103345 Adjacent sequences: A103342 A103343 A103344 this_sequence A103346 A103347 
               A103348
%K A103345 nonn,frac,easy
%O A103345 1,2
%A A103345 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Feb 15 2005