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
Romeo Meštrović, 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.
Maxie 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
a(n)/A007409(n) = zeta(3) + PolyGamma(2, n+1)/2. - Amiram Eldar, Jun 27 2026
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 *)
(* Alternative: *)
Table[Denominator[HarmonicMean[Range[n]^3]], {n, 20}] (* Harvey P. Dale, Aug 20 2017 *)
(* Alternative: *)
Accumulate[1/Range[20]^3]//Numerator (* Harvey P. Dale, Aug 28 2023 *)
PROG
(PARI) a(n)=numerator(sum(k=1, n, 1/k^3)) \\ Charles R Greathouse IV, Jul 19 2011
(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)))
print(list(islice(A007408gen(), 20))) # Michael S. Branicky, Jun 26 2022
CROSSREFS
KEYWORD
nonn,frac,easy,changed
AUTHOR
STATUS
approved