This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A053825 #43 May 04 2025 03:10:50
%S A053825 1,-9,-28,8,-126,252,-344,0,27,1134,-1332,-224,-2198,3096,3528,0,
%T A053825 -4914,-243,-6860,-1008,9632,11988,-12168,0,125,19782,0,-2752,-24390,
%U A053825 -31752,-29792,0,37296,44226,43344,216,-50654,61740,61544,0,-68922,-86688
%N A053825 Dirichlet inverse of sigma_3 function (A001158).
%C A053825 sigma_3(n) is the sum of the cubes of the divisors of n (A001158).
%D A053825 Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 39.
%H A053825 Seiichi Manyama, <a href="/A053825/b053825.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Andrew Howroyd)
%F A053825 Dirichlet g.f.: 1/(zeta(s)*zeta(s-3))
%F A053825 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 A053825 a(n) = Sum_{d|n} mu(n/d)*mu(d)*d^3. - _Ilya Gutkovskiy_, Nov 06 2018
%F A053825 From _Peter Bala_, Jan 17 2024: (Start)
%F A053825 a(n) = Sum_{d divides n} d * A053822(d) * phi(n/d), where the totient function phi(n) = A000010(n).
%F A053825 a(n) = Sum_{d divides n} d^2 * (sigma_1(d))^(-1) * J_2(n/d) and
%F A053825 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 A053825 with(numtheory):seq(add( mobius(n/d)*mobius(d)*d^3, d in divisors(n)), n = 1..100); # _Peter Bala_, Jan 26 2024
%t A053825 a[n_] := Sum[MoebiusMu[n/d] MoebiusMu[d] d^3, {d, Divisors[n]}];
%t A053825 Array[a, 42] (* _Jean-François Alcover_, Aug 16 2019, after _Ilya Gutkovskiy_ *)
%t A053825 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 A053825 (PARI) seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sigma(n, 3)))} \\ _Andrew Howroyd_, Aug 05 2018
%o A053825 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^3*X))[n], ", ")) \\ _Vaclav Kotesovec_, Sep 16 2020
%Y A053825 Cf. A001158, A046692, A053822, A053826, A178448.
%K A053825 sign,mult
%O A053825 1,2
%A A053825 _N. J. A. Sloane_, Apr 08 2000