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

0,3

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

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)

Mathematica

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

Prog

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

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.

Sequence in context: A036094 A133865 A194185 * A096555 A362950 A258533

Adjacent sequences: A282774 A282775 A282776 * A282778 A282779 A282780

Keyword

nonn,easy,mult,changed

Author

Seiichi Manyama, Feb 21 2017

Status

approved