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A370592
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Number of integer partitions of n such that it is possible to choose a different prime factor of each part.
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36
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1, 0, 1, 1, 1, 2, 1, 3, 3, 4, 4, 5, 6, 7, 9, 11, 12, 12, 16, 18, 22, 26, 29, 29, 37, 41, 49, 55, 61, 68, 72, 88, 98, 110, 120, 135, 146, 166, 190, 209, 227, 252, 277, 309, 346, 379, 413, 447, 500, 548, 606, 665, 727, 785, 857, 949, 1033, 1132, 1228, 1328, 1440
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OFFSET
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0,6
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LINKS
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FORMULA
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EXAMPLE
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The partition (10,6,4) has choice (5,3,2) so is counted under a(20).
The a(0) = 1 through a(10) = 4 partitions:
() . (2) (3) (4) (5) (6) (7) (8) (9) (10)
(3,2) (4,3) (5,3) (5,4) (6,4)
(5,2) (6,2) (6,3) (7,3)
(7,2) (5,3,2)
The a(0) = 1 through a(17) = 12 partitions (0 = {}, A..H = 10..17):
0 . 2 3 4 5 6 7 8 9 A B C D E F G H
32 43 53 54 64 65 66 76 86 87 97 98
52 62 63 73 74 75 85 95 96 A6 A7
72 532 83 A2 94 A4 A5 B5 B6
92 543 A3 B3 B4 C4 C5
732 B2 C2 C3 D3 D4
652 653 D2 E2 E3
743 654 754 F2
752 753 763 665
762 853 764
A32 952 A43
B32 7532
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[If[#==1, {}, First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]>0&]], {n, 0, 30}]
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CROSSREFS
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The version for divisors instead of factors is A239312, ranks A368110.
For unlabeled multiset partitions we have A368098, complement A368097.
These partitions have ranks A368100.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A133686, A355739, A355740, A355745, A367771, A367905, A370585, A370586, A370636.
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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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