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

1,5

Comments

The Heinz numbers of these partitions are given by A370348.

Links

Table of n, a(n) for n=1..48.

Example

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)

Mathematica

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

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.

Sequence in context: A241411 A211373 A241734 * A039898 A083690 A241824

Adjacent sequences: A371168 A371169 A371170 * A371172 A371173 A371174

Keyword

nonn

Author

Gus Wiseman, Mar 16 2024

Status

approved