A060046 Generalized sum of divisors function: third diagonal of A060047.

1, 2, 4, 8, 14, 24, 40, 56, 84, 122, 168, 232, 312, 408, 528, 672, 865, 1078, 1336, 1648, 2002, 2424, 2912, 3472, 4116, 4872, 5744, 6648, 7752, 8976, 10304, 11872, 13566, 15424, 17556, 19896, 22414, 25256, 28336, 31584, 35462, 39482, 43728, 48664

Offset: 9

Author

N. J. A. Sloane, Mar 19 2001

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 9..10000
• G. E. Andrews and S. C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms, arXiv:1010.5769 [math.NT], 2010.
• P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

Crossrefs

Cf. A015128.

Programs

Mathematica

nmax = 60; Drop[CoefficientList[Series[-1/3 * Sum[(-1)^k*k*Binomial[k + 2, 5]*x^(k^2), {k, 3, nmax}]/(1 + 2*Sum[(-x)^(k^2), {k, 1, nmax}]), {x, 0, nmax}], x], 9] (* Vaclav Kotesovec, Jul 30 2025 *)

Formula

G.f.: (t(1)^3-3*t(1)*t(2)+2*t(3))/6 where t(i) = Sum((x^(2*n-1)/(1-x^(2*n-1))^2)^i,n=1..inf), i=1..3. - Vladeta Jovovic, Sep 21 2007

G.f.: -(1/3) * ( Sum_{k>=3} (-1)^k * k * binomial(k+2,5) * q^(k^2) ) / ( 1 + 2 * Sum_{k>=1} (-q)^(k^2) ). - Seiichi Manyama, Sep 15 2023

Extensions

More terms from Naohiro Nomoto, Jan 24 2002

More terms from Vladeta Jovovic, Sep 21 2007