A007408 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^3.

1, 9, 251, 2035, 256103, 28567, 9822481, 78708473, 19148110939, 19164113947, 25523438671457, 25535765062457, 56123375845866029, 56140429821090029, 56154295334575853, 449325761325072949, 2207911834254200646437, 245358578943756786493

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• frac

Comments

By Theorem 131 in Hardy and Wright, p^2 divides a(p - 1) for prime p > 5. - T. D. Noe, Sep 05 2002

p^3 divides a(p - 1) for prime p = 37. Primes p such that p divides a((p + 1)/2) are listed in A124787(n) = {3, 11, 17, 89}. - Alexander Adamchuk, Nov 07 2006

a(n)/A007409(n) is the partial sum towards zeta(3), where zeta(s) is the Riemann zeta function. - Alonso del Arte, Dec 30 2012

See the Wolfdieter Lang link under A103345 on Zeta(k, n) with the rationals for k=1..10, g.f.s and polygamma formulas. - Wolfdieter Lang, Dec 03 2013

Denominator of the harmonic mean of the first n cubes. - Colin Barker, Nov 13 2014

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, 1971, page 104.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=1..200
• R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
• Hisanori Mishima, Factorizations of many number sequences
• Hisanori Mishima, Factorizations of many number sequences
• D. Y. Savio, E. A. Lamagna and S.-M. Liu, Summation of harmonic numbers, pp. 12-20 of E. Kaltofen and S. M. Watt, editors, Computers and Mathematics, Springer-Verlag, NY, 1989.
• M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), 10.6.7, Section 4.3.2.

Crossrefs

Cf. A001008, A007406, A007409 (denom), A002117, A124787, A249950.

Programs

Maple

A007408:=n->numer(sum(1/k^3,k=1..n)); map(%,[$1..20]); # M. F. Hasler, Nov 10 2006

Mathematica

Table[Numerator[Sum[1/k^3, {k, n}]], {n, 10}] (* Alonso del Arte, Dec 30 2012 *)
Table[Denominator[HarmonicMean[Range[n]^3]],{n,20}] (* Harvey P. Dale, Aug 20 2017 *)
Accumulate[1/Range[20]^3]//Numerator (* Harvey P. Dale, Aug 28 2023 *)

PARI

a(n)=numerator(sum(k=1,n,1/k^3)) \\ Charles R Greathouse IV, Jul 19 2011

Python

from fractions import Fraction
from itertools import accumulate, count, islice
def A007408gen(): yield from map(lambda x: x.numerator, accumulate(Fraction(1, k**3) for k in count(1)))
print(list(islice(A007408gen(), 20))) # Michael S. Branicky, Jun 26 2022

Formula

Sum_{k = 1 .. n} 1/k^3 = sqrt(sum_{j = 1 .. n} sum_{i = 1 .. n} 1/(i * j)^3). - Alexander Adamchuk, Oct 26 2004