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

0,9

Comments

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

Links

Table of n, a(n) for n=0..55.

Example

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)

Mathematica

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

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.

Sequence in context: A095401 A309511 A195472 * A240538 A326923 A365613

Adjacent sequences: A370803 A370804 A370805 * A370807 A370808 A370809

Keyword

nonn,changed

Author

Gus Wiseman, Mar 04 2024

Extensions

More terms from Jinyuan Wang, Feb 14 2025

Status

approved