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%I M4670 #58 Aug 29 2023 17:20:54
%S 1,9,251,2035,256103,28567,9822481,78708473,19148110939,19164113947,
%T 25523438671457,25535765062457,56123375845866029,56140429821090029,
%U 56154295334575853,449325761325072949,2207911834254200646437,245358578943756786493
%N Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^3.
%C By Theorem 131 in Hardy and Wright, p^2 divides a(p - 1) for prime p > 5. - _T. D. Noe_, Sep 05 2002
%C 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
%C 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
%C 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
%C Denominator of the harmonic mean of the first n cubes. - _Colin Barker_, Nov 13 2014
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, 1971, page 104.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007408/b007408.txt">Table of n, a(n) for n=1..200</a>
%H R. Mestrovic, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha1294.htm">Factorizations of many number sequences</a>
%H D. Y. Savio, E. A. Lamagna and S.-M. Liu, <a href="http://dx.doi.org/10.1007/978-1-4613-9647-5_2">Summation of harmonic numbers</a>, pp. 12-20 of E. Kaltofen and S. M. Watt, editors, Computers and Mathematics, Springer-Verlag, NY, 1989.
%H M. D. Schmidt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Schmidt/multifact.html">Generalized j-Factorial Functions, Polynomials, and Applications </a>, J. Int. Seq. 13 (2010), 10.6.7, Section 4.3.2.
%F 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
%p A007408:=n->numer(sum(1/k^3,k=1..n)); map(%,[$1..20]); # _M. F. Hasler_, Nov 10 2006
%t Table[Numerator[Sum[1/k^3, {k, n}]], {n, 10}] (* _Alonso del Arte_, Dec 30 2012 *)
%t Table[Denominator[HarmonicMean[Range[n]^3]],{n,20}] (* _Harvey P. Dale_, Aug 20 2017 *)
%t Accumulate[1/Range[20]^3]//Numerator (* _Harvey P. Dale_, Aug 28 2023 *)
%o (PARI) a(n)=numerator(sum(k=1,n,1/k^3)) \\ _Charles R Greathouse IV_, Jul 19 2011
%o (Python)
%o from fractions import Fraction
%o from itertools import accumulate, count, islice
%o def A007408gen(): yield from map(lambda x: x.numerator, accumulate(Fraction(1, k**3) for k in count(1)))
%o print(list(islice(A007408gen(), 20))) # _Michael S. Branicky_, Jun 26 2022
%Y Cf. A001008, A007406, A007409, A002117, A124787, A249950.
%K nonn,frac
%O 1,2
%A _N. J. A. Sloane_, _Mira Bernstein_
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