A282548 Expansion of phi_{12, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

0, 1, 4098, 531444, 16785412, 244140630, 2177857512, 13841287208, 68753047560, 282431130813, 1000488301740, 3138428376732, 8920506494928, 23298085122494, 56721594978384, 129747072969720, 281612482805776, 582622237229778, 1157402774071674

Offset: 0

Author

Seiichi Manyama, Feb 18 2017

Keywords

• nonn
• easy
• mult

Comments

Multiplicative because A013959 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}), this sequence (phi_{12, 1}).

Cf. A282549 (E_2*E_4^3), A282576 (E_2*E_6^2), A058550 (E_14).

Cf. A013670.

Programs

Mathematica

Table[n * DivisorSigma[11, n], {n, 0, 18}] (* Amiram Eldar, Sep 06 2023 *)

PARI

a(n) = if(n < 1, 0, n*sigma(n, 11)) \\ Andrew Howroyd, Jul 25 2018

Formula

a(n) = n*A013959(n) for n > 0.

a(n) = (441*A282549(n) + 250*A282576(n) - 691*A058550(n))/65520.

Sum_{k=1..n} a(k) ~ zeta(12) * n^13 / 13. - Amiram Eldar, Sep 06 2023

From Amiram Eldar, Oct 30 2023: (Start)

Multiplicative with a(p^e) = p^e * (p^(11*e+11)-1)/(p^11-1).

Dirichlet g.f.: zeta(s-1)*zeta(s-12). (End)