A220377 Number of partitions of n into three distinct and mutually relatively prime parts.

1, 0, 2, 1, 3, 1, 6, 1, 7, 3, 7, 3, 14, 3, 15, 6, 14, 6, 25, 6, 22, 10, 25, 9, 42, 8, 34, 15, 37, 15, 53, 13, 48, 22, 53, 17, 78, 17, 65, 30, 63, 24, 99, 24, 88, 35, 84, 30, 126, 34, 103, 45, 103, 38, 166, 35, 124, 57, 128, 51, 184, 44, 150, 67, 172, 52, 218

Offset: 6

Author

Carl Najafi, Dec 13 2012

Keywords

• nonn

Comments

The Heinz numbers of these partitions are the intersection of A005117 (strict), A014612 (triples), and A302696 (coprime). - Gus Wiseman, Oct 14 2020

Examples

			For n=10 we have three such partitions: 1+2+7, 1+4+5 and 2+3+5.
From _Gus Wiseman_, Oct 14 2020: (Start)
The a(6) = 1 through a(20) = 15 triples (empty column indicated by dot, A..H = 10..17):
321  .  431  531  532  731  543  751  743  753  754  971  765  B53  875
        521       541       651       752  951  853  B51  873  B71  974
                  721       732       761  B31  871  D31  954  D51  A73
                            741       851       952       972       A91
                            831       941       B32       981       B54
                            921       A31       B41       A71       B72
                                      B21       D21       B43       B81
                                                          B52       C71
                                                          B61       D43
                                                          C51       D52
                                                          D32       D61
                                                          D41       E51
                                                          E31       F41
                                                          F21       G31
                                                                    H21
(End)
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 6..10000 (terms 6..1000 from Seiichi Manyama)

Crossrefs

Cf. A015617, A300815.

A023022 is the 2-part version.

A101271 is the relative prime instead of pairwise coprime version.

A220377*6 is the ordered version.

A305713 counts these partitions of any length, with Heinz numbers A302797.

A307719 is the non-strict version.

A337461 is the non-strict ordered version.

A337563 is the case with no 1's.

A337605 is the pairwise non-coprime instead of pairwise coprime version.

A001399(n-6) counts strict 3-part partitions, with Heinz numbers A007304.

A008284 counts partitions by sum and length, with strict case A008289.

A318717 counts pairwise non-coprime strict partitions.

A326675 ranks pairwise coprime sets.

A327516 counts pairwise coprime partitions.

A337601 counts 3-part partitions whose distinct parts are pairwise coprime.

Cf. A000217, A007360, A023023, A051424, A078374, A087087, A302696, A333227, A337485, A337561.

Programs

Mathematica

Table[Length@Select[ IntegerPartitions[ n, {3}], #[[1]] != #[[2]] != #[[3]] && GCD[#[[1]], #[[2]]] == 1 && GCD[#[[1]], #[[3]]] == 1 && GCD[#[[2]], #[[3]]] == 1 &], {n, 6, 100}]
Table[Count[IntegerPartitions[n,{3}],?(CoprimeQ@@#&&Length[ Union[#]] == 3&)],{n,6,100}] (* _Harvey P. Dale, May 22 2020 *)

PARI

a(n)=my(P=partitions(n));sum(i=1,#P,#P[i]==3&&P[i][1]Charles R Greathouse IV, Dec 14 2012

Formula

a(n > 2) = A307719(n) - 1. - Gus Wiseman, Oct 15 2020