cp's OEIS Frontend

A099828 Numerator of the generalized harmonic number H(n,5) = Sum_{k=1..n} 1/k^5.

%I A099828 #30 Feb 16 2025 08:32:55
%S A099828 1,33,8051,257875,806108207,268736069,4516906311683,144545256245731,
%T A099828 105375212839937899,105376229094957931,16971048697474072945481,
%U A099828 16971114472329088045481,6301272372663207205033976933
%N A099828 Numerator of the generalized harmonic number H(n,5) = Sum_{k=1..n} 1/k^5.
%C A099828 From _Alexander Adamchuk_, Nov 07 2006: (Start)
%C A099828 a(n) is prime for n = {23, 25, 85, 147, 167, ...}.
%C A099828 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 A099828 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 A099828 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 A099828 Alexander Adamchuk, Nov 07 2006, <a href="/A099828/b099828.txt">Table of n, a(n) for n = 1..100</a>
%H A099828 Wolfdieter Lang, <a href="/A103345/a103345.pdf">Rational Zeta(k,n) and more</a>.
%H A099828 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WolstenholmesTheorem.html">Wolstenholme's Theorem</a>.
%F A099828 a(n) = numerator(Sum_{k=1..n} 1/k^5) = numerator(HarmonicNumber[n, 5]).
%e A099828 H(n,5) = {1, 33/32, 8051/7776, 257875/248832, ... } = A099828/A069052.
%e A099828 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 A099828 Numerator[Table[Sum[1/k^5, {k, 1, n}], {n, 1, 20}]]
%t A099828 Numerator[Table[HarmonicNumber[n, 5], {n, 1, 20}]]
%t A099828 Table[Numerator[Sum[1/k^5,{k,1,n}]],{n,1,100}] (* _Alexander Adamchuk_, Nov 07 2006 *)
%o A099828 (PARI) a(n) = numerator(sum(k=1, n, 1/k^5)); \\ _Michel Marcus_, May 10 2020
%Y A099828 Denominators are A069052.
%Y A099828 A099827 = H(n,5) multiplied by (n!)^5.
%Y A099828 Cf. A001008, A007406, A007408, A007410.
%K A099828 nonn,frac
%O A099828 1,2
%A A099828 _Alexander Adamchuk_, Oct 27 2004