A053825 - OEIS

%I #41 Jan 27 2024 10:30:47

%S 1,-9,-28,8,-126,252,-344,0,27,1134,-1332,-224,-2198,3096,3528,0,

%T -4914,-243,-6860,-1008,9632,11988,-12168,0,125,19782,0,-2752,-24390,

%U -31752,-29792,0,37296,44226,43344,216,-50654,61740,61544,0,-68922,-86688

%N Dirichlet inverse of sigma_3 function (A001158).

%C sigma_3(n) is the sum of the cubes of the divisors of n (A001158).

%D Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 39.

%H Seiichi Manyama, <a href="/A053825/b053825.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Andrew Howroyd)

%F Dirichlet g.f.: 1/(zeta(x)*zeta(x-3))

%F 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

%F a(n) = Sum_{d|n} mu(n/d)*mu(d)*d^3. - _Ilya Gutkovskiy_, Nov 06 2018

%F From _Peter Bala_, Jan 17 2024: (Start)

%F a(n) = Sum_{d divides n} d * A053822(d) * phi(n/d), where the totient function phi(n) = A000010(n).

%F a(n) = Sum_{d divides n} d^2 * (sigma_1(d))^(-1) * J_2(n/d) and

%F 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)

%p with(numtheory):seq(add( mobius(n/d)*mobius(d)*d^3, d in divisors(n)), n = 1..100); # _Peter Bala_, Jan 26 2024

%t a[n_] := Sum[MoebiusMu[n/d] MoebiusMu[d] d^3, {d, Divisors[n]}];

%t Array[a, 42] (* _Jean-François Alcover_, Aug 16 2019, after _Ilya Gutkovskiy_ *)

%t 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 *)

%o (PARI) seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sigma(n, 3)))} \\ _Andrew Howroyd_, Aug 05 2018

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^3*X))[n], ", ")) \\ _Vaclav Kotesovec_, Sep 16 2020

%Y Cf. A001158, A046692, A053822, A053826, A178448.

%K sign,mult

%O 1,2

%A _N. J. A. Sloane_, Apr 08 2000