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A330950
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Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.
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27
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1, 1, 1, 2, 2, 3, 3, 7, 7, 11, 11, 22, 15, 30, 42, 77, 42, 101, 56, 176, 176, 231, 135, 490, 490, 490, 792, 1002, 490, 1575, 627, 3010, 2436, 2436, 3718, 5604, 1958, 4565, 6842, 12310, 3718, 14883, 4565, 21637, 26015, 17977, 8349, 53174, 44583, 63261
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OFFSET
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1,4
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COMMENTS
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The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(10) = 11 partitions:
1 11 21 211 32 321 43 5111 522 631
1111 311 2211 421 32111 3222 3331
21111 4111 41111 4221 4321
221111 22221 5311
311111 32211 32221
2111111 222111 33211
11111111 2211111 43111
322111
331111
3211111
31111111
For example, the Heinz number of (3,2) is 15, which is divisible by 5, so (3,2) is counted under a(5).
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Divisible[Times@@Prime/@#, n]&]], {n, 20}]
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CROSSREFS
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The Heinz numbers of these partitions are given by A324851.
Partitions whose product is divisible by their sum are A057568.
Partitions whose Heinz number is divisible by all parts are A330952.
Partitions whose Heinz number is divisible by their product are A324925.
Cf. A056239, A112798, A196050, A324850, A324924, A330953, A330954, A331379, A331381, A331383, A331384.
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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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