A370807 Number of integer partitions of n into parts > 1 such that it is not possible to choose a different prime factor of each part.

0, 0, 0, 0, 1, 0, 3, 1, 4, 4, 8, 9, 15, 17, 25, 30, 43, 54, 72, 87, 115, 139, 181, 224, 283, 342, 429, 519, 647, 779, 967

Offset: 0

Author

Gus Wiseman, Mar 04 2024

Keywords

• nonn
• more

Examples

			The a(0) = 0 through a(11) = 9 partitions:
  .  .  .  .  (22)  .  (33)   (322)  (44)    (333)   (55)     (443)
                       (42)          (332)   (432)   (82)     (533)
                       (222)         (422)   (522)   (433)    (542)
                                     (2222)  (3222)  (442)    (632)
                                                     (622)    (722)
                                                     (3322)   (3332)
                                                     (4222)   (4322)
                                                     (22222)  (5222)
                                                              (32222)
		

Crossrefs

These partitions are ranked by the odd terms of A355529, complement A368100.

The version for set-systems is A367903, complement A367902.

The version for factorizations is A368413, complement A368414.

With ones allowed we have A370593, complement A370592.

For a unique choice we have A370594, ranks A370647.

The version for divisors instead of factors is A370804, complement A370805.

A006530 gives greatest prime factor, least A020639.

A027746 lists prime factors, A112798 indices, length A001222.

A239312 counts condensed partitions, ranks A368110.

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