A103349 Numerators of sum_{k=1..n} 1/k^8 = Zeta(8,n).

1, 257, 1686433, 431733409, 168646292872321, 168646392872321, 972213062238348973121, 248886558707571775009601, 1632944749460578249437992161, 1632944765723715465050248417

Offset: 1

Author

Wolfdieter Lang, Feb 15 2005

Keywords

• nonn
• frac
• easy

Comments

a(n) gives the partial sums, Zeta(8,n) of Euler's Zeta(8). 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 A103350 and for the rationals Zeta(8,n) see the W. Lang link under A103345.

Crossrefs

For k=1..7 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346, A103347/A103348.

Programs

Mathematica

s=0;lst={};Do[s+=n^1/n^9;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 8] // Numerator, {n, 1, 10}] (* Jean-François Alcover, Dec 04 2013 *)
Accumulate[1/Range[10]^8]//Numerator (* Harvey P. Dale, Aug 11 2024 *)

Formula

a(n)=numerator(sum_{k=1..n} 1/k^8).

G.f. for rationals Zeta(8, n): polylogarithm(8, x)/(1-x).