A371130 Number of integer partitions of n such that the number of parts is equal to the number of distinct divisors of parts.

1, 1, 0, 1, 2, 0, 4, 2, 4, 5, 5, 11, 10, 16, 17, 21, 26, 32, 44, 53, 69, 71, 101, 110, 148, 168, 205, 249, 289, 356, 418, 502, 589, 716, 812, 999, 1137, 1365, 1566, 1873, 2158, 2537, 2942, 3449, 4001, 4613, 5380, 6193, 7220, 8224, 9575, 10926, 12683, 14430

Offset: 0

Author

Gus Wiseman, Mar 17 2024

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A370802.

Examples

			The partition (6,2,2,1) has 4 parts and 4 distinct divisors of parts {1,2,3,6} so is counted under a(11).
The a(1) = 1 through a(11) = 11 partitions:
  (1)  .  (21)  (22)  .  (33)   (322)  (71)   (441)   (55)    (533)
                (31)     (51)   (421)  (332)  (522)   (442)   (722)
                         (321)         (422)  (531)   (721)   (731)
                         (411)         (521)  (4311)  (4321)  (911)
                                              (6111)  (6211)  (4322)
                                                              (4331)
                                                              (5321)
                                                              (5411)
                                                              (6221)
                                                              (6311)
                                                              (8111)
		

Crossrefs

The LHS is represented by A001222, distinct A000021.

These partitions are ranked by A370802.

The RHS is represented by A370820, for prime factors A303975.

The strict case is A371128.

For (greater than) instead of (equal to) we have A371171, ranks A370348.

For submultisets instead of parts on the LHS we have A371172.

For (less than) instead of (equal to) we have A371173, ranked by A371168.

Counting only distinct parts on the LHS gives A371178, ranks A371177.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length.

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

Cf. A003963, A319055, A355731, A370803, A370808, A370809, A370813, A370814.

Programs

Mathematica

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