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A330952
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Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by all parts.
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3
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1, 1, 1, 2, 2, 3, 5, 6, 8, 11, 14, 20, 25, 32, 42, 54, 69, 87, 109, 137, 172, 215, 269, 331, 409, 499, 612, 751, 917, 1111, 1344, 1626, 1963, 2359, 2834, 3396, 4065, 4849, 5779, 6865, 8146, 9658, 11424, 13483, 15898, 18710, 21999, 25823, 30272, 35417, 41397
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OFFSET
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0,4
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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(9) = 11 partitions:
1 11 21 211 221 321 2221 3221 621
111 1111 2111 411 3211 4211 3321
11111 2211 4111 22211 22221
21111 22111 32111 32211
111111 211111 41111 42111
1111111 221111 222111
2111111 321111
11111111 411111
2211111
21111111
111111111
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], And@@Table[Divisible[Times@@Prime/@#, i], {i, #}]&]], {n, 0, 30}]
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CROSSREFS
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The Heinz numbers of these partitions are given by A120383.
Partitions whose product is divisible by their sum are A057568.
Partitions whose Heinz number is divisible by their product are A324925.
Partitions whose Heinz number is divisible by their sum are A330950.
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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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