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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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33
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1, 9, 251, 2035, 256103, 28567, 9822481, 78708473, 19148110939, 19164113947, 25523438671457, 25535765062457, 56123375845866029, 56140429821090029, 56154295334575853, 449325761325072949, 2207911834254200646437
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| By Theorem 131 in Hardy and Wright, p^2 divides a(p-1) for prime p > 5. - T. D. Noe (noe(AT)sspectra.com), 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 (alex(AT)kolmogorov.com), Nov 07 2006
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REFERENCES
| G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, 1971, page 104.
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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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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
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FORMULA
| Sum[1/k^3, {k, 1, n}] = Sqrt[Sum[Sum[1/(i*j)^3, {i, 1, n}], {j, 1, n}]] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 26 2004
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MAPLE
| A007408:=n->numer(sum(1/k^3, k=1..n)); map(%, [$1..20]); - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Nov 10 2006
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MATHEMATICA
| s=0; lst={}; Do[s+=n^1/n^4; AppendTo[lst, Numerator[s]], {n, 3*4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 24 2009]
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PROG
| (PARI) a(n)=numerator(sum(k=1, n, 1/k^3)) \\ Charles R Greathouse IV, Jul 19 2011
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CROSSREFS
| Cf. A001008, A007406, A007409, A124787.
Sequence in context: A012202 A012098 A012072 * A066989 A160501 A075987
Adjacent sequences: A007405 A007406 A007407 * A007409 A007410 A007411
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KEYWORD
| nonn,frac
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein
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