OFFSET

1,9

EXAMPLE

The a(7) = 1 through a(15) = 8 partitions (empty column not shown):

43 63 541 83 552 6322 4433 5532

441 4222 3332 6411 7411 7322 6522

222211 5222 62221 44321 84111

33221 63311 333222

65111 432222

72221 3322221

433211 32222211

4322111 333111111

322211111

For example, the partition (3,3,2,2,1) has product 3 * 3 * 2 * 2 * 1 = 36 and sum of primes 5 + 5 + 3 + 3 + 2 = 18, and 36 is divisible by 18, so (3,3,2,2,1) is counted under a(11).

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], Divisible[Times@@#, Plus@@Prime/@#]&]], {n, 30}]

CROSSREFS

The Heinz numbers of these partitions are given by A331378.

Partitions whose product is divisible by their sum are A057568.

Numbers divisible by the sum of their prime indices are A324851.

Partitions whose sum of primes divides their product of primes are A330953.

Partitions whose sum of primes divides of their product are A331381.

Partitions whose product equals their sum of primes are A331383.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 15 2020

STATUS

approved