A370804 Number of non-condensed integer partitions of n into parts > 1.

0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 3, 6, 6, 12, 14, 21, 25, 37, 43, 62, 75, 101, 124, 167, 198, 261, 316, 401, 488, 618, 745, 930, 1119, 1379, 1664, 2032, 2433, 2960, 3537, 4259, 5076, 6094, 7227, 8629, 10205, 12126, 14302, 16932, 19893, 23471, 27502, 32315, 37775

Offset: 0

Author

Gus Wiseman, Mar 03 2024

Keywords

• nonn

Comments

These are partitions without ones such that it is not possible to choose a different divisor of each part.

Examples

			The a(6) = 1 through a(14) = 12 partitions:
  (222)  .  (2222)  (333)   (3322)   (3332)   (3333)    (4333)    (4442)
                    (3222)  (4222)   (5222)   (4422)    (7222)    (5333)
                            (22222)  (32222)  (6222)    (33322)   (5522)
                                              (33222)   (43222)   (8222)
                                              (42222)   (52222)   (33332)
                                              (222222)  (322222)  (43322)
                                                                  (44222)
                                                                  (53222)
                                                                  (62222)
                                                                  (332222)
                                                                  (422222)
                                                                  (2222222)
		

Crossrefs

These partitions have as ranks the odd terms of A355740.

The version with ones is A370320, complement A239312.

The complement without ones is A370805.

The version for prime factors is A370807, with ones A370593.

The version for factorizations is A370813, complement A370814.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

A027746 lists prime factors, A112798 indices, length A001222.

A355731 counts choices of a divisor of each prime index, firsts A355732.

A355741, A355744, A355745 choose prime factors of prime indices.

Cf. A355529, A355739, A367867, A367901, A368110, A368413, A370595, A370806, A370808, A370810.

Programs

Mathematica

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

Extensions

More terms from Jinyuan Wang, Feb 14 2025