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A103345 Numerator of sum_{k=1..n} 1/k^6 = Zeta(6,n). 22
1, 65, 47449, 3037465, 47463376609, 47464376609, 5584183099672241, 357389058474664049, 260537105518334091721, 52107472322919827957, 92311616995117182948130877, 92311647383100199924330877 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For the denominators see A103346. For the rationals Zeta(k,n) for k=1..10, 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.

LINKS

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

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.

FORMULA

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

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-Fran├žois Alcover, Dec 04 2013 *)

CROSSREFS

Cf. A013664. For k=1..5 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052.

Sequence in context: A289946 A219564 A183238 * A291456 A269794 A242283

Adjacent sequences:  A103342 A103343 A103344 * A103346 A103347 A103348

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Feb 15 2005

STATUS

approved

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Last modified February 19 16:02 EST 2019. Contains 320311 sequences. (Running on oeis4.)