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

1, 33, 8051, 257875, 806108207, 268736069, 4516906311683, 144545256245731, 105375212839937899, 105376229094957931, 16971048697474072945481, 16971114472329088045481, 6301272372663207205033976933

Offset: 1

Author

Alexander Adamchuk, Oct 27 2004

Keywords

• nonn
• frac

Comments

From Alexander Adamchuk, Nov 07 2006: (Start)

a(n) is prime for n = {23, 25, 85, 147, 167, ...}.

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.

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)

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

Examples

			H(n,5) = {1, 33/32, 8051/7776, 257875/248832, ... } = A099828/A069052.
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]
		

Links

• Alexander Adamchuk, Nov 07 2006, Table of n, a(n) for n = 1..100
• Wolfdieter Lang, Rational Zeta(k,n) and more.
• Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.

Crossrefs

Denominators are A069052.

A099827 = H(n,5) multiplied by (n!)^5.

Cf. A001008, A007406, A007408, A007410.

Programs

Mathematica

Numerator[Table[Sum[1/k^5, {k, 1, n}], {n, 1, 20}]]
Numerator[Table[HarmonicNumber[n, 5], {n, 1, 20}]]
Table[Numerator[Sum[1/k^5,{k,1,n}]],{n,1,100}] (* Alexander Adamchuk, Nov 07 2006 *)

PARI

a(n) = numerator(sum(k=1, n, 1/k^5)); \\ Michel Marcus, May 10 2020

Formula

a(n) = numerator(Sum_{k=1..n} 1/k^5) = numerator(HarmonicNumber[n, 5]).