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A007408 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^3.
(Formerly M4670)
36
1, 9, 251, 2035, 256103, 28567, 9822481, 78708473, 19148110939, 19164113947, 25523438671457, 25535765062457, 56123375845866029, 56140429821090029, 56154295334575853, 449325761325072949, 2207911834254200646437 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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, 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.

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

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 *)

PROG

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

CROSSREFS

Cf. A001008, A007406, A007409, A002117, A124787, A249950.

Sequence in context: A012202 A012098 A012072 * A066989 A249593 A160501

Adjacent sequences:  A007405 A007406 A007407 * A007409 A007410 A007411

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane, Mira Bernstein

STATUS

approved

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Last modified November 19 10:44 EST 2017. Contains 294936 sequences.