A282777 Expansion of phi_{16, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

0, 1, 65538, 43046724, 4295098372, 152587890630, 2821196197512, 33232930569608, 281483566907400, 1853020317992013, 10000305176108940, 45949729863572172, 184889914172333328, 665416609183179854, 2178019803670969104, 6568408813691796120

Offset: 0

Author

Seiichi Manyama, Feb 21 2017

Keywords

• nonn
• easy
• mult

Comments

Multiplicative because A013963 is. - Andrew Howroyd, Jul 25 2018

References

• George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012. See p. 212.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

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}), A282597 (phi_{14, 1}), this sequence (phi_{16, 1}).

Cf. A282546 (E_2*E_4^4), A282000 (E_4^3*E_6), A282547 (E_2*E_4*E_6^2), A282253 (E_6^3).

Cf. A013674.

Programs

Mathematica

Table[If[n==0, 0, n * DivisorSigma[15, n]], {n, 0, 15}] (* Indranil Ghosh, Mar 11 2017 *)

PARI

for(n=0, 15, print1(if(n==0, 0, n * sigma(n, 15)), ", ")) \\ Indranil Ghosh, Mar 11 2017

Formula

a(n) = n*A013963(n) for n > 0.

a(n) = (2156*A282546(n) - 4156*A282000(n) + 8000*A282547(n)/3 - 2000*A282253(n)/3)/16320.

Sum_{k=1..n} a(k) ~ zeta(16) * n^17 / 17. - Amiram Eldar, Sep 06 2023

From Amiram Eldar, Oct 30 2023: (Start)

Multiplicative with a(p^e) = p^e * (p^(15*e+15)-1)/(p^15-1).

Dirichlet g.f.: zeta(s-1)*zeta(s-16). (End)