A143773 Number of partitions of n such that every part is divisible by number of parts.

1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 5, 1, 4, 3, 6, 1, 8, 1, 7, 5, 6, 1, 14, 2, 7, 8, 11, 1, 17, 1, 14, 11, 9, 3, 29, 1, 10, 15, 23, 1, 28, 1, 23, 25, 12, 1, 51, 2, 20, 25, 32, 1, 44, 11, 39, 31, 15, 1, 94, 1, 16, 40, 52, 19, 64, 1, 57, 45, 44, 1, 126, 1, 19, 83, 74, 6, 90, 1, 124, 63, 21, 1, 186

Offset: 1

Author

Vladeta Jovovic, Aug 31 2008

Keywords

• easy
• nonn
• look

Examples

			The a(18) = 8 partitions are (18), (10 8), (12 6), (14 4), (16 2), (6 6 6), (9 6 3), (12 3 3). - _Gus Wiseman_, Jan 26 2018
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000041, A067538, A143862, A298422, A298423, A298426.

Programs

Mathematica

m = 100;
gf = Sum[x^(k^2)/Product[1-x^(k*i), {i, 1, k}], {k, 1, Sqrt[m]//Ceiling}];
CoefficientList[gf + O[x]^m, x] // Rest (* Jean-François Alcover, May 13 2019 *)

PARI

Vec(sum(k=1,20,x^(k^2)/prod(i=1,k,1-x^(k*i)+O(x^400)))) \\ Max Alekseyev, May 03 2009

Formula

G.f.: Sum(x^(k^2)/Product(1-x^(k*i), i=1..k), k=1..infinity).

For prime p, a(p) = 1 and a(p^2) = 2. For odd prime p, a(2*p) = (p + 1)/2. - Peter Bala, Mar 03 2025

Extensions

More terms from Max Alekseyev, May 03 2009