A103347 Numerator of Sum_{k=1..n} 1/k^7 = zeta(7,n).

1, 129, 282251, 36130315, 2822716691183, 940908897061, 774879868932307123, 99184670126682733619, 650750755630450535274259, 650750820166709327386387, 12681293156341501091194786541177, 12681293507322704937269896541177, 795732373802519483063688931801166184509

Offset

1,2

Comments

a(n) gives the partial sums, zeta(7,n), of Euler's zeta(7). 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 A103348 and for the rationals zeta(7,n) see the W. Lang link under A103345.

Links

Robert Israel, Table of n, a(n) for n = 1..336

Formula

a(n) = numerator(Sum_{k=1..n} 1/k^7).

G.f. for rationals zeta(7, n): polylogarithm(7, x)/(1-x).

From Amiram Eldar, Jun 27 2026: (Start)

a(n)/A103348(n) = zeta(7) + PolyGamma(6, n+1)/6!.

Limit_{n->oo} a(n)/A103348(n) = zeta(7) (A013665). (End)

Maple

f:= n -> numer(Psi(6, n+1)/720 + Zeta(7)):
map(f, [$1..20]); # Robert Israel, Mar 28 2018

Mathematica

s=0; lst={}; Do[s+=n^1/n^8; AppendTo[lst, Numerator[s]], {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
(* Alternative: *)
Table[ HarmonicNumber[n, 7] // Numerator, {n, 1, 12}] (* Jean-François Alcover, Dec 04 2013 *)

Crossrefs

Cf. A103348 (denominators), A013665.

For k=1..6 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346.

Sequence in context: A242229 A351137 A138586 * A291505 A275097 A337809

Adjacent sequences: A103344 A103345 A103346 * A103348 A103349 A103350

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Feb 15 2005

Status

approved