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For unlabeled multiset partitions we have A370646, cf. A368098, A368097.

The partition (10,6,4) has unique choice (5,3,2) so is counted under a(20).

The a(0) = 1 through a(16) = 11 partitions (0 = (), A..G = 10..16):

0 . 2 3 4 5 . 7 8 9 64 B 66 D 86 87 G

32 43 53 54 73 74 75 85 95 96 97

52 62 63 532 83 A2 94 A4 A5 B5

72 92 543 B2 B3 B4 C4

732 652 C2 C3 D3

653 D2 E2

743 654 754

752 753 763

762 853

A32 952

B32

Cf. A000040, A000041, A000720, A003963, A355739, `A355740, `A355744, `A355745, `A367905, `A368110.

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

nonn,more,changed

allocated for Gus WisemanNumber of integer partitions of n such that only one set can be obtained by choosing a different prime factor of each part.

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

0,6

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)

The a(0) = 1 through a(16) = 11 partitions (0 = (), A..G = 10..16):

0 . 2 3 4 5 . 7 8 9 64 B 66 D 86 87 G

32 43 53 54 73 74 75 85 95 96 97

52 62 63 532 83 A2 94 A4 A5 B5

72 92 543 B2 B3 B4 C4

732 652 C2 C3 D3

653 D2 E2

743 654 754

752 753 763

762 853

A32 952

B32

Table[Length[Select[IntegerPartitions[n], Length[Union[Sort/@Select[Tuples[If[#==1, {}, First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]]==1&]], {n, 0, 30}]

The version for set-systems is A367904, ranks A367908.

Multisets of this type are ranked by A368101, cf. A368100, A355529.

The version for subsets is A370584, cf. A370582, A370583, A370586, A370587.

Maximal sets of this type are counted by A370585.

For existence we have A370592.

For nonexistence we have A370593.

For divisors instead of factors we have A370595.

For subsets and binary indices we have A370638, cf. A370636, A370637.

The version for factorizations is A370645, cf. A368414, A368413.

For unlabeled multiset partitions we have A370646, cf. A368098, A368097.

These partitions have ranks A370647.

A006530 gives greatest prime factor, least A020639.

A027746 lists prime factors, A112798 indices, length A001222.

A355741 counts ways to choose a prime factor of each prime index.

Cf. A000040, A000041, A000720, A003963, A355739, `A355740, `A355744, `A355745, `A367905, `A368110.

allocated

nonn,more

Gus Wiseman, Feb 29 2024

approved

editing

allocated for Gus Wiseman

allocated

approved