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

1,4

Links

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

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A305974 A332267 A161148 * A323524 A354713 A265893

Adjacent sequences: A143770 A143771 A143772 * A143774 A143775 A143776

Keyword

easy,nonn,look

Author

Vladeta Jovovic, Aug 31 2008

Extensions

More terms from Max Alekseyev, May 03 2009

Status

approved