A389719 Number of partitions of n into parts where each part divides the largest part, and each part occurs at most twice if it is the largest part, and at most once otherwise.

1, 2, 2, 3, 3, 4, 4, 4, 5, 6, 6, 7, 6, 6, 9, 8, 8, 11, 9, 9, 13, 13, 12, 13, 12, 11, 15, 15, 13, 19, 18, 14, 18, 18, 21, 26, 22, 21, 25, 23, 22, 27, 22, 21, 31, 30, 29, 33, 29, 30, 38, 35, 32, 40, 42, 39, 44, 41, 36, 45, 41, 36, 48, 42, 44, 58, 50, 48, 58, 59, 55, 63, 55, 49, 64

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Formula

G.f.: Sum_{k>=1} q^k * Product_{d|k} (1+q^d).

Example

a(9) = 5 counts these partitions: 9, 81, 63, 621, 441.

Mathematica

Nmax=80; Table[SeriesCoefficient[Sum[q^k*Product[If[Mod[k, d]==0, 1+q^d, 1], {d, 1, k}], {k, 1, Nmax}], {q, 0, n}], {n, 1, Nmax}] (* Vincenzo Librandi, Mar 09 2026 *)

Prog

(PARI) my(N=80, q='q+O('q^N)); Vec(sum(k=1, N, q^k*prod(d=1, k, if(k%d==0, 1+q^d, 1))))
(Magma) N := 80; R<q> := PowerSeriesRing(Integers(), N+1); S := &+[ q^k * &*[1 + q^d : d in Divisors(k)] : k in [1..N] ]; [Coefficient(S, n) : n in [1..N]]; // Vincenzo Librandi, Mar 09 2026

Crossrefs

Cf. A000009, A130689, A343347.

Sequence in context: A132921 A255232 A181988 * A194173 A028825 A396205

Adjacent sequences: A389716 A389717 A389718 * A389720 A389721 A389722

Keyword

nonn

Author

Seiichi Manyama, Mar 09 2026

Status

approved