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

0,3

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.

Links

Table of n, a(n) for n=0..60.

Example

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

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}]

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.

Sequence in context: A076640 A326198 A324105 * A169819 A134681 A218703

Adjacent sequences: A328868 A328869 A328870 * A328872 A328873 A328874

Keyword

nonn

Author

Gus Wiseman, Nov 12 2019

Status

approved