A282597 Expansion of phi_{14, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

0, 1, 16386, 4782972, 268468228, 6103515630, 78373779192, 678223072856, 4398583447560, 22876806803877, 100012207113180, 379749833583252, 1284076017413616, 3937376385699302, 11113363271818416, 29192944359852360, 72066391204823056, 168377826559400946

Offset: 0

Author

Seiichi Manyama, Feb 19 2017

Keywords

• nonn
• easy
• mult

Comments

Multiplicative because A013961 is. - Andrew Howroyd, Jul 25 2018

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), A282254 (phi_{10, 1}), A282548 (phi_{12, 1}), this sequence (phi_{14, 1}).

Cf. A282012 (E_4^4), A282287 (E_4*E_6^2), A282596 (E_2*E_4^2*E_6).

Cf. A013672.

Programs

Mathematica

Table[n * DivisorSigma[13, n], {n, 0, 17}] (* Amiram Eldar, Sep 06 2023 *)

PARI

a(n) = if(n < 1, 0, n*sigma(n, 13)) \\ Andrew Howroyd, Jul 25 2018

Formula

a(n) = n*A013961(n) for n > 0.

a(n) = (3*A282012(n) + 4*A282287(n) - 7*A282596(n))/144.

Sum_{k=1..n} a(k) ~ zeta(14) * n^15 / 15. - Amiram Eldar, Sep 06 2023

From Amiram Eldar, Oct 30 2023: (Start)

Multiplicative with a(p^e) = p^e * (p^(13*e+13)-1)/(p^13-1).

Dirichlet g.f.: zeta(s-1)*zeta(s-14). (End)