A371132 Number of integer partitions of n with fewer distinct parts than distinct divisors of parts.

0, 0, 1, 1, 2, 3, 5, 6, 10, 14, 21, 28, 40, 53, 73, 96, 130, 170, 223, 288, 375, 480, 616, 780, 990, 1245, 1567, 1954, 2440, 3024, 3745, 4610, 5674, 6947, 8499, 10349, 12591, 15258, 18468, 22277, 26841, 32238, 38673, 46262, 55278, 65881, 78423, 93136, 110477

Offset

0,5

Comments

The Heinz numbers of these partitions are given by A371179.

Links

Table of n, a(n) for n=0..48.

Example

The partition (4,3,1,1) has 3 distinct parts {1,3,4} and 4 distinct divisors of parts {1,2,3,4}, so is counted under a(9).
The a(0) = 0 through a(9) = 14 partitions:
  .  .  (2)  (3)  (4)   (5)   (6)    (7)     (8)      (9)
                  (22)  (32)  (33)   (43)    (44)     (54)
                        (41)  (42)   (52)    (53)     (63)
                              (222)  (61)    (62)     (72)
                              (411)  (322)   (332)    (81)
                                     (4111)  (422)    (333)
                                             (431)    (432)
                                             (611)    (441)
                                             (2222)   (522)
                                             (41111)  (621)
                                                      (3222)
                                                      (4311)
                                                      (6111)
                                                      (411111)

Mathematica

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

Crossrefs

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

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

The complement counting all parts on the LHS is A371172, ranks A371165.

Counting all parts on the LHS gives A371173, ranks A371168.

The complement is counted by A371178, ranks A371177.

These partitions are ranked by A371179.

The strict case is A371180, complement A371128.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length.

Cf. A003963, A239312, A319055, A355740, A370802, A370803, A370808, A370813, A371130, A371171.

Sequence in context: A280013 A039840 A039845 * A347868 A039848 A018494

Adjacent sequences: A371129 A371130 A371131 * A371133 A371134 A371135

Keyword

nonn

Author

Gus Wiseman, Mar 17 2024

Status

approved