OFFSET
1,2
COMMENTS
sigma_3(n) is the sum of the cubes of the divisors of n (A001158).
REFERENCES
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 39.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)
FORMULA
Dirichlet g.f.: 1/(zeta(x)*zeta(x-3))
Multiplicative with a(p^1) = -1-p^3, a(p^2) = p^3, a(p^e) = 0 for e>=3. - Mitch Harris, Jun 27 2005
a(n) = Sum_{d|n} mu(n/d)*mu(d)*d^3. - Ilya Gutkovskiy, Nov 06 2018
From Peter Bala, Jan 17 2024: (Start)
a(n) = Sum_{d divides n} d^2 * (sigma_1(d))^(-1) * J_2(n/d) and
a(n) = Sum_{d divides n} d^3 * (sigma_k(d))^(-1) * J_(k+3)(n/d), where (sigma_k(n))^(-1) denotes the Dirichlet inverse of the divisor sum function sigma_k(n) and J_k(n) denotes the Jordan totient function. (End)
MAPLE
with(numtheory):seq(add( mobius(n/d)*mobius(d)*d^3, d in divisors(n)), n = 1..100); # Peter Bala, Jan 26 2024
MATHEMATICA
a[n_] := Sum[MoebiusMu[n/d] MoebiusMu[d] d^3, {d, Divisors[n]}];
Array[a, 42] (* Jean-François Alcover, Aug 16 2019, after Ilya Gutkovskiy *)
f[p_, e_] := If[e == 1, -p^3 - 1, If[e == 2, p^3, 0]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)
PROG
(PARI) seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sigma(n, 3)))} \\ Andrew Howroyd, Aug 05 2018
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^3*X))[n], ", ")) \\ Vaclav Kotesovec, Sep 16 2020
CROSSREFS
KEYWORD
sign,mult
AUTHOR
N. J. A. Sloane, Apr 08 2000
STATUS
approved