A318750 a(n) = Sum_{k=1..n} k * tau_3(k), where tau_3 is A007425.

1, 7, 16, 40, 55, 109, 130, 210, 264, 354, 387, 603, 642, 768, 903, 1143, 1194, 1518, 1575, 1935, 2124, 2322, 2391, 3111, 3261, 3495, 3765, 4269, 4356, 5166, 5259, 5931, 6228, 6534, 6849, 8145, 8256, 8598, 8949, 10149, 10272, 11406, 11535, 12327, 13137, 13551

Offset: 1

Author

Vaclav Kotesovec, Sep 02 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..100000
• Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

Crossrefs

Cf. A007425, A034718, A061201, A318413, A318414.

Cf. A001620, A082633.

Programs

Mathematica

Accumulate[Table[n*Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, 100}]]
(* Asymptotics: *) n^2 * (Log[n]^2 + (6*EulerGamma - 1)*Log[n] + 6*EulerGamma^2 - 3*EulerGamma - 6*StieltjesGamma[1] + 1/2) / 4 (* Vaclav Kotesovec, Sep 09 2018 *)

PARI

tau_3(k) = vecprod(apply(e -> (e+1)*(e+2)/2, factor(k)[, 2]));
a(n) = sum(k = 1, n,  k * tau_3(k)); \\ Amiram Eldar, Jan 18 2025

Formula

a(n) = Sum_{k=1..n} A034718(k).

a(n) ~ n^2 * (log(n)^2 + (6*g-1)*log(n) + 6*g^2 - 3*g - 6*g1 + 1/2) / 4, where g is the Euler-Mascheroni constant A001620 and g1 is the first Stieltjes constant A082633. - Vaclav Kotesovec, Sep 09 2018