A370806 Number of non-strict condensed integer partitions of n.

0, 0, 0, 0, 1, 0, 1, 1, 3, 2, 4, 4, 8, 9, 11, 14, 19, 24, 29, 39, 47, 58, 70, 85, 104, 129, 152, 184, 223, 264, 313, 374, 442, 524, 617, 719, 852, 993, 1159, 1344, 1579, 1817, 2114, 2440, 2826, 3250, 3750, 4297, 4944, 5662, 6475, 7404, 8462, 9634, 10972, 12480

Offset: 0

Author

Gus Wiseman, Mar 04 2024

Keywords

• nonn

Comments

These are non-strict partitions such that it is possible to choose a different divisor of each part.

Examples

			The a(4) = 1 through a(13) = 9 partitions:
  (22)  .  (33)  (322)  (44)   (441)  (55)   (443)   (66)    (544)
                        (332)  (522)  (433)  (533)   (444)   (553)
                        (422)         (442)  (722)   (552)   (661)
                                      (622)  (4322)  (633)   (733)
                                                     (822)   (922)
                                                     (4332)  (4432)
                                                     (4431)  (5332)
                                                     (5322)  (5422)
                                                             (6322)
		

Crossrefs

This is the non-strict case of A239312, complement A370320.

These partitions have as ranks the nonsquarefree terms of A368110.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

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

A370592 counts factor-choosable partitions, complement A370593.

A370814 counts condensed factorizations, complement A370813.

Cf. A355739, A355740, A370595, A370804, A370808, A370810.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],!UnsameQ@@# && Length[Select[Tuples[Divisors/@#],UnsameQ@@#&]]>0&]],{n,0,30}]

Extensions

More terms from Jinyuan Wang, Feb 14 2025