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

0,4

Comments

The Heinz numbers of these partitions are given by A371288.

Links

Table of n, a(n) for n=0..56.

Example

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)

Mathematica

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

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.

Sequence in context: A161389 A266118 A287802 * A214207 A302497 A044051

Adjacent sequences: A371281 A371282 A371283 * A371285 A371286 A371287

Keyword

nonn

Author

Gus Wiseman, Mar 22 2024

Status

approved