A328871 Number of integer partitions of n whose distinct parts are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 2, 6, 2, 7, 5, 7, 2, 10, 2, 11, 7, 14, 2, 16, 4, 19, 8, 22, 2, 30, 3, 29, 14, 37, 8, 48, 4, 50, 19, 59, 5, 82, 4, 81, 28, 93, 8, 128, 9, 128, 38, 147, 8, 199, 19, 196, 52, 223, 12, 308

Offset: 0

Author

Gus Wiseman, Nov 12 2019

Keywords

• nonn

Comments

A partition with no two distinct parts divisible is said to be stable, and a partition with no two distinct parts relatively prime is said to be intersecting, so these are just stable intersecting partitions.

Examples

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

Crossrefs

The Heinz numbers of these partitions are A329366.

Replacing "intersecting" with "relatively prime" gives A328676.

Stable partitions are A305148.

Intersecting partitions are A328673.

Cf. A000837, A285573, A303362, A305148, A316476, A328671, A328677.

Programs

Mathematica

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],stableQ[Union[#],Divisible]&&stableQ[Union[#],GCD[#1,#2]==1&]&]],{n,0,30}]