A002127 - OEIS

A002127

MacMahon's generalized sum of divisors function.
(Formerly M2770 N1114)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,2

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

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

George E. Andrews and Simon 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.

Simon Rose, What literature is known about MacMahon's generalized sum-of-divisors function?

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

EXAMPLE

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

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 *)

PROG

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

CROSSREFS

A diagonal of A060043.

Cf. A002128.