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A007408
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Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^3.
(Formerly M4670)
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39
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1, 9, 251, 2035, 256103, 28567, 9822481, 78708473, 19148110939, 19164113947, 25523438671457, 25535765062457, 56123375845866029, 56140429821090029, 56154295334575853, 449325761325072949, 2207911834254200646437, 245358578943756786493
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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.
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FORMULA
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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
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MAPLE
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MATHEMATICA
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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 *)
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PROG
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(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)))
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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