A124787 -id:A124787 - cp's OEIS frontend

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

N. J. A. Sloane, Mira Bernstein

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

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).

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.

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

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

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 *)

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

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

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

A124876 Number of prime factors (counted with multiplicity) in factorization of A007408(n).

Original entry on oeis.org

0, 2, 1, 3, 2, 4, 3, 2, 3, 5, 3, 3, 3, 2, 1, 4, 2, 5, 1, 3, 2, 6, 2, 4, 2, 1, 3, 5, 3, 6, 1, 2, 3, 2, 3, 10, 4, 4, 5, 5, 8, 7, 7, 2, 4, 7, 3, 2, 4, 3, 2, 5, 3, 4, 2, 8, 3, 4, 4, 5, 3, 3, 7, 2, 5, 10, 4, 2, 6, 8, 3, 6, 6, 4, 3, 6, 4, 7, 4, 4, 3, 4, 8, 5, 7, 4

Offset: 1

M. F. Hasler, Nov 11 2006

nonn

			a(1) = 0 since A007408(1) = 1 contains no prime factor,
a(2) = 2 since A007408(2) = 9 = 3 * 3,
a(3) = 1 since A007408(3) = 251 is prime,
a(6) = 4 since A007408(6) = 7 * 7 * 11 * 53.

Amiram Eldar, Table of n, a(n) for n = 1..106

Cf. A001222, A007408, A124877, A124787.

seq( add(op(2,j),j=op(2,(ifactors@A007408)(n))), n=1..28 );
A001222 := proc(n) numtheory[bigomega](n) ; end: b := fscanf("b007408.txt","%d %d") : while b <> [] do printf("%d, ",A001222(op(2,b))) ; b := fscanf("b007408.txt","%d %d") : od : # R. J. Mathar, May 18 2007

Table[PrimeOmega[Numerator[Sum[1/k^3, {k, 1, n}]]], {n, 1, 50}] (* Amiram Eldar, Feb 09 2020 *)
PrimeOmega[Numerator[Accumulate[1/Range[50]^3]]] (* Harvey P. Dale, Aug 28 2023 *)

a(n) = A001222(A007408(n)). - R. J. Mathar, May 18 2007

More terms from R. J. Mathar, May 18 2007

More terms from Amiram Eldar, Feb 09 2020

cp's OEIS Frontend

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

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