A338317 Number of integer partitions of n with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime.

1, 0, 1, 1, 2, 2, 3, 4, 5, 6, 7, 11, 11, 16, 16, 19, 25, 32, 34, 44, 46, 53, 66, 80, 88, 101, 116, 132, 150, 180, 204, 229, 254, 287, 331, 366, 426, 473, 525, 584, 662, 742, 835, 922, 1013, 1128, 1262, 1408, 1555, 1711, 1894, 2080, 2297, 2555, 2806, 3064, 3376

Offset: 0

Author

Gus Wiseman, Oct 24 2020

Keywords

• nonn

Examples

			The a(2) = 1 through a(12) = 11 partitions (A = 10, B = 11, C = 12):
  2   3   4    5    6     7     8      9      A       B       C
          22   32   33    43    44     54     55      65      66
                    222   52    53     72     73      74      75
                          322   332    333    433     83      444
                                2222   522    532     92      543
                                       3222   3322    443     552
                                              22222   533     732
                                                      722     3333
                                                      3332    5322
                                                      5222    33222
                                                      32222   222222
		

Crossrefs

A007359 (A302568) gives the strict case.

A101268 (A335235) gives pairwise coprime or singleton compositions.

A200976 (A338318) gives the pairwise non-coprime instead of coprime version.

A304709 (A304711) gives partitions whose distinct parts are pairwise coprime, with strict case A305713 (A302797).

A304712 (A338331) allows 1's, with strict version A007360 (A302798).

A327516 (A302696) gives pairwise coprime partitions.

A328673 (A328867) gives partitions with no distinct relatively prime parts.

A338315 (A337987) does not consider singletons coprime.

A338317 (A338316) gives these partitions.

A337462 (A333227) gives pairwise coprime compositions.

A337485 (A337984) gives pairwise coprime integer partitions with no 1's.

A337665 (A333228) gives compositions with pairwise coprime distinct parts.

A337667 (A337666) gives pairwise non-coprime compositions.

A337697 (A022340 /\ A333227) = pairwise coprime compositions with no 1's.

A337983 (A337696) gives pairwise non-coprime strict compositions, with unordered version A318717 (A318719).

Cf. A051424, A289508, A302569, A303140, A337694.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&(SameQ@@#||CoprimeQ@@Union[#])&]],{n,0,15}]

Formula

The Heinz numbers of these partitions are given by A338316. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.