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A370592
Number of integer partitions of n such that it is possible to choose a different prime factor of each part.
36
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
OFFSET
0,6
FORMULA
a(n) = A000041(n) - A370593(n).
EXAMPLE
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
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[If[#==1, {}, First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]>0&]], {n, 0, 30}]
CROSSREFS
The version for divisors instead of factors is A239312, ranks A368110.
The version for set-systems is A367902, ranks A367906, unlabeled A368095.
The complement for set-systems is A367903, ranks A367907, unlabeled A368094.
For unlabeled multiset partitions we have A368098, complement A368097.
These partitions have ranks A368100.
The version for factorizations is A368414, complement A368413.
The complement is counted by A370593, ranks A355529.
For a unique choice we have A370594, ranks A370647.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts choices of a prime factor of each prime index.
Sequence in context: A342016 A360923 A123621 * A151662 A049786 A282611
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 29 2024
STATUS
approved