A371284 Number of integer partitions of n whose distinct parts form the set of divisors of some number.

0, 1, 1, 2, 3, 4, 5, 8, 9, 11, 12, 16, 18, 23, 25, 32, 36, 42, 47, 57, 62, 73, 81, 96, 106, 123, 132, 154, 168, 190, 207, 240, 259, 293, 317, 359, 388, 434, 469, 529, 574, 635, 688, 768, 826, 915, 987, 1093, 1181, 1302, 1397, 1540, 1662, 1818, 1959, 2149, 2309

Offset: 0

Author

Gus Wiseman, Mar 22 2024

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A371288.

Examples

			The partition y = (10,5,5,5,2,2,1) has distinct parts {1,2,5,10}, which form the set of divisors of 10, so y is counted under a(30).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (11)  (21)   (31)    (221)    (51)      (331)      (71)
             (111)  (211)   (311)    (2211)    (421)      (3311)
                    (1111)  (2111)   (3111)    (511)      (4211)
                            (11111)  (21111)   (2221)     (5111)
                                     (111111)  (22111)    (22211)
                                               (31111)    (221111)
                                               (211111)   (311111)
                                               (1111111)  (2111111)
                                                          (11111111)
		

Crossrefs

The strict case is A054973, ranks A371283 (unsorted version A275700).

These partitions have ranks A371288.

A000005 counts divisors, row-lengths of A027750.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

Cf. A001221, A002865, A239312, A370803, A371172, A371286, A371285.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], Union[#]==Divisors[Max[#]]&]],{n,0,30}]