A103716 Numerators of sum_{k=1..n} 1/k^10 =: Zeta(10,n).

1, 1025, 60526249, 61978938025, 605263128567754849, 605263138567754849, 170971856382109814342232401, 175075181098169912564190119249, 10338014371627802833957102351534201, 413520574906423083987893722912609

Offset: 1

Author

Wolfdieter Lang, Feb 15 2005

Keywords

• nonn
• frac
• easy

Comments

a(n) gives the partial sums, Zeta(10,n), of Euler's Zeta(10). Zeta(k,n) is also called H(k,n) because for k=1 these are the harmonic numbers H(n) = A001008/A002805.

For the denominators see A103717 and for the rationals Zeta(10,n) see the W. Lang link under A103345.

Crossrefs

For k=1..9 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346, A103347/A103348, A103349/A103350, A103351/A103352.

Programs

Mathematica

s=0;lst={};Do[s+=n^1/n^11;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 10] // Numerator, {n, 1, 10}] (* Jean-François Alcover, Dec 04 2013 *)

Formula

a(n) = numerator(sum_{k=1..n} 1/k^10).

G.f. for rationals Zeta(10, n): polylogarithm(10, x)/(1-x).