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A353396
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Number of integer partitions of n whose Heinz number has prime shadow equal to the product of prime shadows of its parts.
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4
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1, 0, 1, 1, 0, 2, 0, 3, 1, 3, 4, 3, 7, 5, 9, 8, 12, 15, 15, 20, 21, 25, 31, 33, 38, 42, 46, 56, 61, 67, 78, 76, 96, 100, 114, 131, 130, 157, 157, 185, 200, 214, 236, 253, 275, 302, 333, 351, 386, 408, 440, 486, 515, 564, 596, 633, 691, 734, 800, 854, 899, 964
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OFFSET
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0,6
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COMMENTS
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The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.
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LINKS
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EXAMPLE
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The a(8) = 1 through a(14) = 9 partitions (A..D = 10..13):
(53) (72) (73) (B) (75) (D) (B3)
(621) (532) (A1) (651) (B2) (752)
(4221) (631) (4331) (732) (A21) (761)
(4411) (6321) (43321) (A31)
(6411) (44311) (C11)
(43221) (6521)
(44211) (9221)
(54221)
(64211)
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MATHEMATICA
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red[n_]:=If[n==1, 1, Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n], Times@@red/@#==red[Times@@Prime/@#]&]], {n, 0, 15}]
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CROSSREFS
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The RHS (product of prime shadows) is A353394, first appearances A353397.
These partitions are ranked by A353395.
A003963 gives product of prime indices.
A324850 lists numbers divisible by the product of their prime indices.
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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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