A338315 Number of integer partitions of n with no 1's whose distinct parts are pairwise coprime, where a singleton is not considered coprime unless it is (1).

0, 0, 0, 0, 0, 1, 0, 3, 2, 4, 4, 10, 6, 15, 13, 16, 21, 31, 29, 43, 41, 50, 63, 79, 81, 99, 113, 129, 145, 179, 197, 228, 249, 284, 328, 363, 418, 472, 522, 581, 655, 741, 828, 921, 1008, 1123, 1259, 1407, 1546, 1709, 1889, 2077, 2292, 2554, 2799, 3061, 3369

Offset: 0

Author

Gus Wiseman, Oct 23 2020

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A337987. 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.

Examples

			The a(5) = 1 through a(13) = 15 partitions (empty column indicated by dot, A = 10, B = 11):
  32   .  43    53    54     73     65      75      76
          52    332   72     433    74      543     85
          322         522    532    83      552     94
                      3222   3322   92      732     A3
                                    443     5322    B2
                                    533     33222   544
                                    722             553
                                    3332            733
                                    5222            922
                                    32222           4333
                                                    5332
                                                    7222
                                                    33322
                                                    52222
                                                    322222
		

Crossrefs

A200976 is a pairwise non-coprime instead of pairwise coprime version.

A304709 allows 1's, with strict case A305713 and Heinz numbers A304711.

A318717 counts pairwise non-coprime strict partitions.

A337485 is the strict version, with Heinz numbers A337984.

A337987 gives the Heinz numbers of these partitions.

A338317 considers singletons coprime, with Heinz numbers A338316.

A007359 counts singleton or pairwise coprime partitions with no 1's.

A327516 counts pairwise coprime partitions, ranked by A302696.

A328673 counts partitions with no two distinct parts relatively prime.

A337462 counts pairwise coprime compositions, ranked by A333227.

A337561 counts pairwise coprime strict compositions.

A337665 counts compositions whose distinct parts are pairwise coprime.

A337667 counts pairwise non-coprime compositions, ranked by A337666.

A337697 counts pairwise coprime compositions with no 1's.

Cf. A051424, A101268, A220377, A302568, A328867, A333228, A337563.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&CoprimeQ@@Union[#]&]],{n,0,30}]