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A370594
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Number of integer partitions of n such that only one set can be obtained by choosing a different prime factor of each part.
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16
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1, 0, 1, 1, 1, 2, 0, 3, 3, 4, 3, 4, 5, 5, 8, 10, 11, 7, 14, 13, 19, 23, 24, 20, 30, 33, 40, 47, 49, 55, 53, 72, 80, 90, 92, 110, 110, 132, 154, 169, 180, 201, 218, 246, 281, 302, 323, 348, 396, 433, 482, 530, 584, 618, 670, 754, 823, 903, 980, 1047, 1137
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OFFSET
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0,6
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LINKS
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EXAMPLE
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The partition (10,6,4) has unique choice (5,3,2) so is counted under a(20).
The a(0) = 1 through a(12) = 5 partitions:
() . (2) (3) (4) (5) . (7) (8) (9) (6,4) (11) (6,6)
(3,2) (4,3) (5,3) (5,4) (7,3) (7,4) (7,5)
(5,2) (6,2) (6,3) (5,3,2) (8,3) (10,2)
(7,2) (9,2) (5,4,3)
(7,3,2)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Length[Union[Sort/@Select[Tuples[If[#==1, {}, First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]]==1&]], {n, 0, 30}]
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CROSSREFS
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Maximal sets of this type are counted by A370585.
For divisors instead of factors we have A370595.
These partitions have ranks A370647.
A355741 counts ways to choose a prime factor of each prime index.
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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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