

A330954


Number of integer partitions of n whose product is divisible by the sum of primes of their parts.


16



0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 4, 2, 3, 9, 8, 18, 15, 25, 35, 44, 50, 70, 71, 93, 141, 158, 226, 286, 337, 439, 532, 648, 789, 1013, 1261, 1454, 1776, 2176, 2701, 3258, 3823, 4606, 5521, 6613, 7810, 9202, 11074, 13145, 15498, 18413, 21818, 25774, 30481, 35718
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,9


LINKS

Table of n, a(n) for n=1..54.


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.
Cf. A000040, A001414, A036844, A056239, A324850, A326149, A330950, A331379, A331382, A331384, A331415, A331416.
Sequence in context: A256443 A046068 A166281 * A107468 A023632 A286844
Adjacent sequences: A330951 A330952 A330953 * A330955 A330956 A330957


KEYWORD

nonn


AUTHOR

Gus Wiseman, Jan 15 2020


STATUS

approved



