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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

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Last modified September 26 01:25 EDT 2020. Contains 337346 sequences. (Running on oeis4.)