A370593 Number of integer partitions of n such that it is not possible to choose a different prime factor of each part.

0, 1, 1, 2, 4, 5, 10, 12, 19, 26, 38, 51, 71, 94, 126, 165, 219, 285, 369, 472, 605, 766, 973, 1226, 1538, 1917, 2387, 2955, 3657, 4497, 5532, 6754, 8251, 10033, 12190, 14748, 17831, 21471, 25825, 30976, 37111, 44331, 52897, 62952, 74829, 88755, 105145, 124307

Offset: 0

Author

Gus Wiseman, Feb 29 2024

Keywords

• nonn

Examples

			The a(0) = 0 through a(7) = 12 partitions:
  .  (1)  (11)  (21)   (22)    (41)     (33)      (61)
                (111)  (31)    (221)    (42)      (322)
                       (211)   (311)    (51)      (331)
                       (1111)  (2111)   (222)     (421)
                               (11111)  (321)     (511)
                                        (411)     (2221)
                                        (2211)    (3211)
                                        (3111)    (4111)
                                        (21111)   (22111)
                                        (111111)  (31111)
                                                  (211111)
                                                  (1111111)
		

Crossrefs

The complement for divisors instead of factors is A239312, ranks A368110.

These partitions have ranks A355529, complement A368100.

The complement for set-systems is A367902, ranks A367906, unlabeled A368095.

The version for set-systems is A367903, ranks A367907, unlabeled A368094.

For unlabeled multiset partitions we have A368097, complement A368098.

The version for factorizations is A368413, complement A368414.

The complement is counted by A370592.

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.

Cf. A000040, A000720, A133686, A355739, A355740, A367771, A367867, A367905, A370583, A370585, A370586, A370636.

Programs

Mathematica

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

Formula

a(n) = A000041(n) - A370592(n).