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

0,6

Comments

The Heinz numbers of these partitions are given by A371168.

Links

Table of n, a(n) for n=0..52.

Example

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)

Mathematica

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

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.

Sequence in context: A099258 A264905 A254052 * A105746 A166309 A349472

Adjacent sequences: A371170 A371171 A371172 * A371174 A371175 A371176

Keyword

nonn

Author

Gus Wiseman, Mar 16 2024

Status

approved