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

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..100000

Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

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

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 *)

Prog

(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

Crossrefs

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

Cf. A001620, A082633.

Sequence in context: A286710 A036834 A020941 * A024625 A055553 A066009

Adjacent sequences: A318747 A318748 A318749 * A318751 A318752 A318753

Keyword

nonn,changed

Author

Vaclav Kotesovec, Sep 02 2018

Status

approved