A328673 Number of integer partitions of n in which no two distinct parts are relatively prime.

1, 1, 2, 2, 3, 2, 5, 2, 6, 4, 9, 2, 15, 2, 17, 10, 23, 2, 39, 2, 46, 18, 58, 2, 95, 8, 103, 31, 139, 2, 219, 3, 232, 59, 299, 22, 452, 4, 492, 104, 645, 5, 920, 5, 1006, 204, 1258, 8, 1785, 21, 1994, 302, 2442, 11, 3366, 71, 3738, 497, 4570, 18, 6253, 24, 6849

Offset: 0

Author

Gus Wiseman, Oct 29 2019

Keywords

• nonn

Comments

A partition with no two distinct parts relatively prime is said to be intersecting.

Examples

			The a(1) = 1 through a(10) = 9 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        63         55
              1111         42               62        333        64
                           222              422       111111111  82
                           111111           2222                 442
                                            11111111             622
                                                                 4222
                                                                 22222
                                                                 1111111111
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350

Crossrefs

The Heinz numbers of these partitions are A328867 (strict case is A318719).

The relatively prime case is A328672.

The strict case is A318717.

The version for non-isomorphic multiset partitions is A319752.

The version for set-systems is A305843.

The version involving all parts (not just distinct ones) is A200976.

Cf. A000837, A202425, A305148, A305854, A306006, A316476, A326910.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],And@@(GCD[##]>1&)@@@Subsets[Union[#],{2}]&]],{n,0,20}]

Formula

a(n > 0) = A200976(n) + 1.