A379813 a(n) = sigma_1(n) * sigma_3(n).

1, 27, 112, 511, 756, 3024, 2752, 8775, 9841, 20412, 15984, 57232, 30772, 74304, 84672, 145111, 88452, 265707, 137200, 386316, 308224, 431568, 292032, 982800, 488281, 830844, 817600, 1406272, 731700, 2286144, 953344, 2359287, 1790208, 2388204, 2080512, 5028751

Offset: 1

Author

Amiram Eldar, Jan 03 2025

Keywords

• nonn
• easy
• mult

Comments

See A379812 for more links and Ramanujan's general formula.

References

• Srinivasa Ramanujan, Collected papers, edited by G. H. Hardy et al., Chelsea, 1962, chapter 17, pp. 133-135.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Srinivasa Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Mathematics, Vol. 45 (1916), pp. 81-84.

Crossrefs

Cf. A000203, A001158, A013663, A072861, A091258, A091259, A092348, A092345, A356533, A356534, A379812, A379814.

Programs

Mathematica

a[n_] := Times @@ DivisorSigma[{1, 3}, n]; Array[a, 50]

PARI

a(n) = {my(f = factor(n)); sigma(f) * sigma(f, 3);}

Formula

a(n) = A000203(n) * A001158(n).

Multiplicative with a(p^e) = (p^(e+1)-1) * (p^(3*e+3)-1) / ((p-1) * (p^3-1)).

Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-3) * zeta(s-4) / zeta(2*s-4).

Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 7 * zeta(5) / 4 = 1.81462357150089737107... .