A103349 Numerator of Sum_{k=1..n} 1/k^8 = zeta(8,n).

1, 257, 1686433, 431733409, 168646292872321, 168646392872321, 972213062238348973121, 248886558707571775009601, 1632944749460578249437992161, 1632944765723715465050248417, 350036214341656523811791485040141377, 350036215152423104696224989140141377

Offset

1,2

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.

Links

Table of n, a(n) for n=1..12.

Formula

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

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

From Amiram Eldar, Jun 27 2026: (Start)

a(n)/A103350(n) = zeta(8) - PolyGamma(7, n+1)/7!.

Limit_{n->oo} a(n)/A103350(n) = zeta(8) (A013666). (End)

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 *)
(* Alternative: *)
Table[ HarmonicNumber[n, 8] // Numerator, {n, 1, 10}] (* Jean-François Alcover, Dec 04 2013 *)
(* Alternative: *)
Accumulate[1/Range[10]^8]//Numerator (* Harvey P. Dale, Aug 11 2024 *)

Crossrefs

Cf. A103350 (denominators), A013666.

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

Sequence in context: A218723 A097736 A283510 * A291506 A275098 A194155

Adjacent sequences: A103346 A103347 A103348 * A103350 A103351 A103352

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Feb 15 2005

Extensions

More terms from Amiram Eldar, Jun 27 2026

Status

approved