A371128 Number of strict integer partitions of n containing all distinct divisors of all parts.

1, 1, 0, 1, 1, 0, 2, 1, 2, 1, 2, 2, 3, 3, 3, 5, 3, 5, 6, 7, 7, 8, 8, 9, 12, 13, 13, 14, 15, 16, 19, 23, 25, 26, 26, 27, 36, 37, 40, 42, 46, 50, 55, 66, 65, 71, 71, 82, 90, 102, 103, 114, 117, 130, 147, 154, 166, 176, 182, 194, 228, 239, 259, 267, 287, 307, 336

Offset: 0

Author

Gus Wiseman, Mar 18 2024

Keywords

• nonn

Comments

Also strict integer partitions such that the number of parts is equal to the number of distinct divisors of all parts.

Examples

			The a(9) = 1 through a(19) = 7 partitions (A..H = 10..17):
  531  721   731   B1    751   D1    B31    D21    B51    H1     B71
       4321  5321  5421  931   B21   7521   7531   D31    9531   D51
                   6321  7321  7421  8421   64321  B321   A521   B521
                                     9321          65321  B421   D321
                                     54321         74321  75321  75421
                                                          84321  76321
                                                                 94321
		

Crossrefs

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

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

Strict case of A371130 (ranks A370802) and A371178 (ranks A371177).

The complement is counted by A371180, non-strict A371132.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length.

A305148 counts partitions without divisors, strict A303362, ranks A316476.

Cf. A000837, A003963, A239312, A285573, A319055, A370803, A370808, A371171, A371172, A371173.

Programs

Mathematica

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