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%I #29 May 31 2022 06:48:25
%S 1,33,8051,257875,806108207,268736069,4516906311683,144545256245731,
%T 105375212839937899,105376229094957931,16971048697474072945481,
%U 16971114472329088045481,6301272372663207205033976933
%N Numerator of the generalized harmonic number H(n,5) = Sum_{k=1..n} 1/k^5.
%C From _Alexander Adamchuk_, Nov 07 2006: (Start)
%C a(n) is prime for n = {23, 25, 85, 147, 167, ...}.
%C There is a Wolstenholme-like theorem: p divides a(p-1) for prime p and p^2 divides a(p-1) for prime p > 7.
%C Also, p^3 divides a(p-1) for prime p = 5; p divides a((p-1)/2) for prime p = 37; p divides a((p-1)/3) for prime p = 37; p divides a((p-1)/4) for prime p = 37; p divides a((p-1)/5) for prime p = 11; p^2 divides a((p-1)/6) for prime p = 37; p divides a((p+1)/4) for prime p = 83; p divides a((p+1)/5) for prime p = 29; and p divides a((p+1)/6) for prime p = 11. (End)
%C See the Wolfdieter Lang link for information about Zeta(k, n) = H(n, k) with the rationals for k = 1..10, g.f.s, and polygamma formulas. - _Wolfdieter Lang_, Dec 03 2013
%H Alexander Adamchuk, Nov 07 2006, <a href="/A099828/b099828.txt">Table of n, a(n) for n = 1..100</a>
%H Wolfdieter Lang, <a href="/A103345/a103345.pdf">Rational Zeta(k,n) and more</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WolstenholmesTheorem.html">Wolstenholme's Theorem</a>.
%F a(n) = numerator(Sum_{k=1..n} 1/k^5) = numerator(HarmonicNumber[n, 5]).
%e H(n,5) = {1, 33/32, 8051/7776, 257875/248832, ... } = A099828/A069052.
%e For example, a(2) = numerator(1 + 1/2^5) = numerator(33/32) = 33 and a(3) = numerator(1 + 1/2^5 + 1/3^5) = numerator(8051/7776) = 8051. [Edited by _Petros Hadjicostas_, May 10 2020]
%t Numerator[Table[Sum[1/k^5, {k, 1, n}], {n, 1, 20}]]
%t Numerator[Table[HarmonicNumber[n, 5], {n, 1, 20}]]
%t Table[Numerator[Sum[1/k^5,{k,1,n}]],{n,1,100}] (* _Alexander Adamchuk_, Nov 07 2006 *)
%o (PARI) a(n) = numerator(sum(k=1, n, 1/k^5)); \\ _Michel Marcus_, May 10 2020
%Y Denominators are A069052.
%Y A099827 = H(n,5) multiplied by (n!)^5.
%Y Cf. A001008, A007406, A007408, A007410.
%K nonn,frac
%O 1,2
%A _Alexander Adamchuk_, Oct 27 2004
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