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%I #9 Mar 01 2024 15:54:56
%S 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,
%T 61,68,72,88,98,110,120,135,146,166,190,209,227,252,277,309,346,379,
%U 413,447,500,548,606,665,727,785,857,949,1033,1132,1228,1328,1440
%N Number of integer partitions of n such that it is possible to choose a different prime factor of each part.
%F a(n) = A000041(n) - A370593(n).
%e The partition (10,6,4) has choice (5,3,2) so is counted under a(20).
%e The a(0) = 1 through a(10) = 4 partitions:
%e () . (2) (3) (4) (5) (6) (7) (8) (9) (10)
%e (3,2) (4,3) (5,3) (5,4) (6,4)
%e (5,2) (6,2) (6,3) (7,3)
%e (7,2) (5,3,2)
%e The a(0) = 1 through a(17) = 12 partitions (0 = {}, A..H = 10..17):
%e 0 . 2 3 4 5 6 7 8 9 A B C D E F G H
%e 32 43 53 54 64 65 66 76 86 87 97 98
%e 52 62 63 73 74 75 85 95 96 A6 A7
%e 72 532 83 A2 94 A4 A5 B5 B6
%e 92 543 A3 B3 B4 C4 C5
%e 732 B2 C2 C3 D3 D4
%e 652 653 D2 E2 E3
%e 743 654 754 F2
%e 752 753 763 665
%e 762 853 764
%e A32 952 A43
%e B32 7532
%t Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]>0&]],{n,0,30}]
%Y The version for divisors instead of factors is A239312, ranks A368110.
%Y The version for set-systems is A367902, ranks A367906, unlabeled A368095.
%Y The complement for set-systems is A367903, ranks A367907, unlabeled A368094.
%Y For unlabeled multiset partitions we have A368098, complement A368097.
%Y These partitions have ranks A368100.
%Y The version for factorizations is A368414, complement A368413.
%Y The complement is counted by A370593, ranks A355529.
%Y For a unique choice we have A370594, ranks A370647.
%Y A006530 gives greatest prime factor, least A020639.
%Y A027746 lists prime factors, A112798 indices, length A001222.
%Y A355741 counts choices of a prime factor of each prime index.
%Y Cf. A000040, A000720, A133686, A355739, A355740, A355745, A367771, A367905, A370585, A370586, A370636.
%K nonn
%O 0,6
%A _Gus Wiseman_, Feb 29 2024
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