A337603 Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is not considered coprime unless it is (1).

0, 0, 0, 1, 3, 6, 9, 9, 18, 15, 24, 21, 42, 24, 51, 30, 54, 42, 93, 45, 102, 54, 99, 69, 162, 66, 150, 87, 168, 96, 264, 93, 228, 120, 246, 126, 336, 132, 315, 168, 342, 162, 486, 165, 420, 216, 411, 213, 618, 207, 558, 258, 540, 258, 783, 264, 654, 324, 660

Offset: 0

Author

Gus Wiseman, Sep 20 2020

Keywords

• nonn

Examples

			The a(3) = 1 through a(8) = 18 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)
           (1,2,1)  (1,2,2)  (1,2,3)  (1,3,3)  (1,2,5)
           (2,1,1)  (1,3,1)  (1,3,2)  (1,5,1)  (1,3,4)
                    (2,1,2)  (1,4,1)  (2,2,3)  (1,4,3)
                    (2,2,1)  (2,1,3)  (2,3,2)  (1,5,2)
                    (3,1,1)  (2,3,1)  (3,1,3)  (1,6,1)
                             (3,1,2)  (3,2,2)  (2,1,5)
                             (3,2,1)  (3,3,1)  (2,3,3)
                             (4,1,1)  (5,1,1)  (2,5,1)
                                               (3,1,4)
                                               (3,2,3)
                                               (3,3,2)
                                               (3,4,1)
                                               (4,1,3)
                                               (4,3,1)
                                               (5,1,2)
                                               (5,2,1)
                                               (6,1,1)
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

Crossrefs

A014311 intersected with A333228 ranks these compositions.

A220377*6 is the strict case.

A337461 is the strict case except for any number of 1's.

A337601 is the unordered version.

A337602 considers all singletons to be coprime.

A337665 counts these compositions of any length, ranked by A333228 with complement A335238.

A000217(n - 2) counts 3-part compositions.

A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.

A007318 and A097805 count compositions by length.

A051424 counts pairwise coprime or singleton partitions.

A101268 counts pairwise coprime or singleton compositions.

A304711 ranks partitions whose distinct parts are pairwise coprime.

A305713 counts strict pairwise coprime partitions.

A327516 counts pairwise coprime partitions, with strict case A305713.

A333227 ranks pairwise coprime compositions.

Cf. A000740, A001840, A007359, A087087, A178472, A284825, A302696, A307719, A335235, A337561, A337695.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],CoprimeQ@@Union[#]&]],{n,0,100}]