A371172 Number of integer partitions of n with as many submultisets as distinct divisors of parts.

0, 0, 1, 1, 0, 1, 0, 3, 2, 3, 1, 4, 2, 1, 2, 3, 4, 2, 4, 1, 5, 2, 7, 5, 9, 4, 9, 15, 18, 16, 24, 13, 17, 23, 23, 22, 34, 17, 30, 31, 36, 29, 43, 21, 30, 35, 44, 28, 47, 19, 44

Offset: 0

Author

Gus Wiseman, Mar 16 2024

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A371165.

Examples

			The partition (8,6,6) has 6 submultisets {(8,6,6),(8,6),(6,6),(8),(6),()} and 6 distinct divisors of parts {1,2,3,4,6,8}, so is counted under a(20).
The a(17) = 2 through a(24) = 9 partitions:
  (17)    (9,9)     (19)  (11,9)    (14,7)  (13,9)    (23)       (21,3)
  (13,4)  (15,3)          (15,5)    (17,4)  (21,1)    (19,4)     (22,2)
          (6,6,6)         (8,6,6)           (8,8,6)   (22,1)     (8,8,8)
          (12,3,3)        (12,4,4)          (10,6,6)  (15,4,4)   (10,8,6)
                          (18,1,1)          (16,3,3)  (12,10,1)  (12,6,6)
                                            (18,2,2)             (12,7,5)
                                            (20,1,1)             (18,3,3)
                                                                 (20,2,2)
                                                                 (12,10,2)
		

Crossrefs

The RHS is represented by A370820.

Counting parts on the LHS gives A371130 (ranks A370802), strict A371128.

These partitions are ranked by A371165.

A000005 counts divisors.

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

Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).

Cf. A003963, A319055, A355739, A370803, A370808, A370813, A370814, A371166.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], Length[Divisors[Times@@Prime/@#]] == Length[Union@@Divisors/@#]&]],{n,0,30}]