A002127 MacMahon's generalized sum of divisors function.

1, 3, 9, 15, 30, 45, 67, 99, 135, 175, 231, 306, 354, 465, 540, 681, 765, 945, 1040, 1305, 1386, 1695, 1779, 2205, 2290, 2754, 2835, 3438, 3480, 4185, 4272, 5076, 5004, 6100, 5985, 7155, 7154, 8325, 8190, 9840, 9471, 11241, 11055, 12870, 12420, 14911

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of partitions of n with two designated summands. For example: a(5) = 9 because there are 9 partitions of 5 with two designated summands: [4'+ 1'], [3'+ 2'], [3'+ 1'+ 1], [3'+ 1 + 1'], [2'+ 2 + 1'], [2 + 2'+ 1'], [2'+ 1'+ 1 + 1], [2'+ 1 + 1'+ 1], [2'+ 1 + 1 + 1']. - Omar E. Pol, Jul 23 2025

Examples

			x^3 + 3*x^4 + 9*x^5 + 15*x^6 + 30*x^7 + 45*x^8 + 67*x^9 + 99*x^10 + ...
		

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

• John Cerkan, Table of n, a(n) for n = 3..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.
• William Craig, Jan-Willem van Ittersum and Ken Ono, Integer partitions detect the primes, PNAS, Vol. 121, No. 39 (2024), e2409417121.
• P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113; Coll. Papers II, pp. 303-341.
• S. Rose, What literature is known about MacMahon's generalized sum-of-divisors function?

Crossrefs

A diagonal of A060043.

Cf. A002128.

Column 2 of A385001.

Programs

Mathematica

A002127[n_] := (DivisorSigma[3, n] - (2*n - 1)*DivisorSigma[1, n])/8;
Array[A002127, 50, 3] (* Paolo Xausa, Jul 04 2025, after Michael Somos's PARI *)

PARI

{a(n) = if( n<1, 0, ( sigma( n, 3) - (2*n - 1) * sigma(n) ) / 8)} /* Michael Somos, Jan 10 2012 */

Formula

G.f.: (Sum_{k>=0} (-1)^k * (2*k + 1) * binomial( k+2, 4) * x^( k*(k+1) / 2 )) / (5 * Sum_{k>=0} (-1)^k * (2*k + 1) * x^( k*(k+1) / 2 )). - Michael Somos, Jan 10 2012

a(n) = (n^2 - 3*n + 2) * A000203(n) / 8 iff n is an odd prime (see Craig link et al.).

Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / (4!*5!). - Vaclav Kotesovec, Aug 01 2025

Extensions

More terms from Vladeta Jovovic, Nov 11 2001