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

0,3

Comments

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

Links

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

Formula

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

Example

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

Mathematica

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

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.

Sequence in context: A363724 A345268 A164941 * A115119 A066656 A164896

Adjacent sequences: A328670 A328671 A328672 * A328674 A328675 A328676

Keyword

nonn

Author

Gus Wiseman, Oct 29 2019

Status

approved