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A330954
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Number of integer partitions of n whose product is divisible by the sum of primes of their parts.
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16
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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
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OFFSET
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1,9
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LINKS
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EXAMPLE
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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).
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Divisible[Times@@#, Plus@@Prime/@#]&]], {n, 30}]
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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