A002132 Generalized sum of divisors function.

1, 2, 4, 8, 14, 18, 28, 40, 52, 70, 88, 104, 140, 168, 196, 240, 278, 320, 380, 440, 504, 562, 644, 720, 808, 910, 1000, 1120, 1240, 1360, 1488, 1600, 1789, 1938, 2100, 2296, 2452, 2660, 2880, 3080, 3292, 3542, 3784, 4048, 4400, 4572, 4868, 5280, 5502, 5850

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 4..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

A diagonal of A060047.

Cf. A015128.

Programs

Mathematica

nmax = 60; Drop[CoefficientList[Series[1/2 * Sum[(-1)^k*k*Binomial[k + 1, 3]*x^(k^2), {k, 2, nmax}]/(1 + 2*Sum[(-x)^(k^2), {k, 1, nmax}]), {x, 0, nmax}], x], 4] (* Vaclav Kotesovec, Jul 30 2025 *)

Formula

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

Extensions

More terms from Naohiro Nomoto and Vladeta Jovovic, Jan 25 2002