A002134 Generalized divisor function. Number of partitions of n with exactly three part sizes.

1, 2, 5, 10, 15, 25, 37, 52, 67, 97, 117, 154, 184, 235, 277, 338, 385, 469, 531, 630, 698, 810, 910, 1038, 1144, 1295, 1425, 1577, 1741, 1938, 2089, 2301, 2505, 2700, 2970, 3189, 3444, 3703, 4004, 4242, 4617, 4882, 5244, 5558, 5999, 6221, 6755, 7050, 7576

Offset: 6

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Examples

			a(8) = 5 because we have 5+2+1, 4+3+1, 4+2+1+1, 3+2+2+1, 3+2+1+1+1.
		

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

• Alois P. Heinz, Table of n, a(n) for n = 6..10000
• 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 A060177.

Column k=3 of A116608. - Alois P. Heinz, Nov 07 2012

Programs

Maple

# Using function P from A365676:
A002134 := n -> P(n, 3, n): seq(A002134(n), n = 6..54); # Peter Luschny, Sep 15 2023

Mathematica

nn=40;sss=Sum[Sum[Sum[x^(i+j+k)/(1-x^i)/(1-x^j)/(1-x^k),{k,1,j-1}], {j,1,i-1}], {i,1,nn}]; Drop[CoefficientList[Series[sss,{x,0,nn}],x],6]  (* Geoffrey Critzer, Sep 13 2012 *)

Formula

G.f.: Sum_{i>=1} Sum_{j=1..i-1} Sum_{k=1..j-1} x^(i+j+k)/((1-x^i)*(1-x^j)* (1-x^k)). - Geoffrey Critzer, Sep 13 2012

Extensions

Better description and more terms from Naohiro Nomoto, Jan 24 2002

More terms from Vladeta Jovovic, Nov 02 2003