A103345 Numerator of Sum_{k=1..n} 1/k^6 = Zeta(6,n).

1, 65, 47449, 3037465, 47463376609, 47464376609, 5584183099672241, 357389058474664049, 260537105518334091721, 52107472322919827957, 92311616995117182948130877, 92311647383100199924330877, 445570781131605573859221176881493, 445570839299219762020391212081493

Offset: 1

Author

Wolfdieter Lang, Feb 15 2005

Keywords

• nonn
• frac
• easy

Comments

For the rationals Zeta(k,n) for k = 1..10 and n = 1..20, see the W. Lang link.

a(n) gives the partial sum, Zeta(6,n), of Euler's (later Riemann's) Zeta(6). Zeta(k,n), k >= 2, is sometimes also called H(k,n) because for k = 1 these would be the harmonic numbers A001008/A002805. However, H(1,n) does not give partial sums of a convergent series.

Examples

			The first few fractions are 1, 65/64, 47449/46656, 3037465/2985984, 47463376609/46656000000, ... = A103345/A103346. - _Petros Hadjicostas_, May 10 2020
		

Links

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Wolfdieter Lang, Rational Zeta(k,n) and more.

Crossrefs

Cf. A013664, A291456. For the denominators, see A103346.

For k=1..5, see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052.

Programs

Mathematica

s=0; lst={}; Do[s+=n^1/n^7; AppendTo[lst,Numerator[s]],{n,3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 6] // Numerator, {n, 1, 12}] (* Jean-François Alcover, Dec 04 2013 *)

Formula

a(n) = numerator(Sum_{k=1..n} 1/k^6) = numerator(A291456(n)/(n!)^6).

G.f. for rationals Zeta(6, n): polylogarithm(6, x)/(1-x).

Zeta(6, n) = (1/945)*Pi^6 - psi(5, n+1)/5!, see eq. 6.4.3 with m = 5, p. 260, of the Abramowitz-Stegun reference. - Wolfdieter Lang, Dec 03 2013