A370594 Number 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, 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

Offset: 0

Author

Gus Wiseman, Feb 29 2024

Keywords

• nonn

Examples

			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)
		

Crossrefs

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.

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, A367905, A368110.

Programs

Mathematica

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