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A099827 Generalized harmonic number H(n,5) = Sum_{k=1..n} 1/k^5 multiplied by (n!)^5.

%I A099827 #37 Feb 16 2025 08:32:55
%S A099827 0,1,33,8051,8252000,25795462624,200610400564224,3371852494046112768,
%T A099827 110492114540967125581824,6524555433591956305924325376,
%U A099827 652461835742417609568446054400000,105080260346474296336209157187174400000
%N A099827 Generalized harmonic number H(n,5) = Sum_{k=1..n} 1/k^5 multiplied by (n!)^5.
%C A099827 Note that a(n) is divisible by n, except when n is prime. Also, a(n+1) is divisible by n, except when n is prime or n = 0.
%H A099827 Seiichi Manyama, <a href="/A099827/b099827.txt">Table of n, a(n) for n = 0..120</a>
%H A099827 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%F A099827 a(n) = (n!)^5 * Sum_{k=1..n} 1/k^5 = (n!)^5 * HarmonicNumber[n, 5] = (n!)^5 * A099828(n)/A069052(n).
%F A099827 a(0) = 0, a(1) = 1, a(n+1) = (n^5 + (n+1)^5)*a(n) - n^10*a(n-1) for n > 0. - _Seiichi Manyama_, Aug 24 2017
%F A099827 a(n) ~ Zeta(5) * (2*Pi)^(5/2) * n^(5*n+5/2) / exp(5*n). - _Vaclav Kotesovec_, Aug 27 2017
%F A099827 Sum_{n>=0} a(n) * x^n / (n!)^5 = polylog(5,x) / (1 - x). - _Ilya Gutkovskiy_, Jul 14 2020
%e A099827 a(2) = (2!)^5 * (1 + 1/2^5) = 2^5 + 1 = 33,
%e A099827 a(3) = (3!)^5 * (1 + 1/2^5 + 1/3^5) = 6^5 + 3^5 + 1 = 8051.
%t A099827 Table[(n!)^5*Sum[1/k^5, {k, 1, n}], {n, 0, 13}] or Table[(n!)^5*HarmonicNumber[n, 5], {n, 0, 13}]
%Y A099827 Cf. A001008, A001819, A008515, A069052, A099828.
%Y A099827 Column k = 5 of A291556.
%K A099827 nonn
%O A099827 0,3
%A A099827 _Alexander Adamchuk_, Oct 27 2004
%E A099827 a(0) = 0 prepended by _Seiichi Manyama_, Aug 23 2017
%E A099827 Name edited by _Petros Hadjicostas_, May 10 2020