A371171 Number of integer partitions of n with more parts than distinct divisors of parts.

0, 0, 1, 1, 2, 4, 5, 9, 12, 18, 26, 34, 50, 65, 92, 121, 161, 209, 274, 353, 456, 590, 745, 950, 1195, 1507, 1885, 2350, 2923, 3611, 4465, 5485, 6735, 8223, 10050, 12195, 14822, 17909, 21653, 26047, 31340, 37557, 44990, 53708, 64068, 76241, 90583, 107418

Offset: 1

Author

Gus Wiseman, Mar 16 2024

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A370348.

Examples

			The partition (3,2,1,1) has 4 parts {1,2,3,4} and 3 distinct divisors of parts {1,2,3}, so is counted under a(7).
The a(0) = 0 through a(8) = 12 partitions:
  .  .  (11)  (111)  (211)   (221)    (222)     (331)      (2222)
                     (1111)  (311)    (2211)    (511)      (3221)
                             (2111)   (3111)    (2221)     (3311)
                             (11111)  (21111)   (3211)     (4211)
                                      (111111)  (4111)     (5111)
                                                (22111)    (22211)
                                                (31111)    (32111)
                                                (211111)   (41111)
                                                (1111111)  (221111)
                                                           (311111)
                                                           (2111111)
                                                           (11111111)
		

Crossrefs

The partitions are ranked by A370348.

The opposite version is A371173, ranked by A371168.

The RHS is represented by A370820, positions of twos A371127.

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

For submultisets instead of parts on the LHS we get ranks A371167.

A000005 counts divisors.

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}]