A371173 Number of integer partitions of n with fewer parts than distinct divisors of parts.

0, 0, 1, 1, 1, 3, 2, 4, 6, 7, 11, 11, 17, 20, 26, 34, 44, 56, 67, 84, 102, 131, 156, 195, 232, 283, 346, 411, 506, 598, 721, 855, 1025, 1204, 1448, 1689, 2018, 2363, 2796, 3265, 3840, 4489, 5242, 6104, 7106, 8280, 9595, 11143, 12862, 14926, 17197, 19862, 22841

Offset: 0

Author

Gus Wiseman, Mar 16 2024

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A371168.

Examples

			The partition (4,3,2) has 3 parts {2,3,4} and 4 distinct divisors of parts {1,2,3,4}, so is counted under a(9).
The a(2) = 1 through a(10) = 11 partitions:
  (2)  (3)  (4)  (5)    (6)    (7)    (8)      (9)      (10)
                 (3,2)  (4,2)  (4,3)  (4,4)    (5,4)    (6,4)
                 (4,1)         (5,2)  (5,3)    (6,3)    (7,3)
                               (6,1)  (6,2)    (7,2)    (8,2)
                                      (4,3,1)  (8,1)    (9,1)
                                      (6,1,1)  (4,3,2)  (4,3,3)
                                               (6,2,1)  (5,3,2)
                                                        (5,4,1)
                                                        (6,2,2)
                                                        (6,3,1)
                                                        (8,1,1)
		

Crossrefs

The RHS is represented by A370820.

The version for equality is A371130 (ranks A370802), strict A371128.

For submultisets instead of parts on the LHS we get ranks A371166.

These partitions are ranked by A371168.

The opposite version is A371171, ranks A370348.

A000005 counts divisors.

A355731 counts choices of a divisor of each prime index, firsts A355732.

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

Cf. A003963, A319055, A355739, A370803, A370808, A370809, A370813, A370814.

Programs

Mathematica

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