A371178 Number of integer partitions of n containing all divisors of all parts.

1, 1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 37, 48, 62, 80, 101, 127, 162, 202, 252, 312, 386, 475, 585, 713, 869, 1056, 1278, 1541, 1859, 2232, 2675, 3196, 3811, 4534, 5386, 6379, 7547, 8908, 10497, 12345, 14501, 16999, 19897, 23253, 27135, 31618, 36796, 42756

Offset: 0

Author

Gus Wiseman, Mar 17 2024

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A371177.

Also partitions such that the number of distinct parts is equal to the number of distinct divisors of parts.

Examples

			The partition (4,2,1,1) contains all distinct divisors {1,2,4}, so is counted under a(8).
The partition (4,4,3,2,2,2,1) contains all distinct divisors {1,2,3,4} so is counted under 4 + 4 + 3 + 2 + 2 + 2 + 1 = 18. - _David A. Corneth_, Mar 18 2024
The a(0) = 1 through a(8) = 12 partitions:
  ()  (1)  (11)  (21)   (31)    (221)    (51)      (331)      (71)
                 (111)  (211)   (311)    (321)     (421)      (521)
                        (1111)  (2111)   (2211)    (511)      (3221)
                                (11111)  (3111)    (2221)     (3311)
                                         (21111)   (3211)     (4211)
                                         (111111)  (22111)    (5111)
                                                   (31111)    (22211)
                                                   (211111)   (32111)
                                                   (1111111)  (221111)
                                                              (311111)
                                                              (2111111)
                                                              (11111111)
		

Crossrefs

The LHS is represented by A001221, distinct case of A001222.

For partitions with no divisors of parts we have A305148, ranks A316476.

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

The strict case is A371128.

Counting all parts on the LHS gives A371130, ranks A370802.

The complement is counted by A371132.

For submultisets instead of distinct parts we have A371172, ranks A371165.

These partitions have ranks A371177.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length.

Cf. A000837, A003963, A239312, A285573, A305148, A319055, A355529, A370803, A370808, A370813, A371168, A371171, A371173.

Programs

Mathematica

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