A337602 Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

0, 0, 0, 1, 3, 6, 10, 9, 18, 16, 24, 21, 43, 24, 51, 31, 54, 42, 94, 45, 102, 55, 99, 69, 163, 66, 150, 88, 168, 96, 265, 93, 228, 121, 246, 126, 337, 132, 315, 169, 342, 162, 487, 165, 420, 217, 411, 213, 619, 207, 558, 259, 540, 258, 784, 264, 654, 325, 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)  (2,2,2)  (3,1,3)  (1,6,1)
                             (2,3,1)  (3,2,2)  (2,1,5)
                             (3,1,2)  (3,3,1)  (2,3,3)
                             (3,2,1)  (5,1,1)  (2,5,1)
                             (4,1,1)           (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

The complement in A014311 of A337695 ranks these compositions.

A220377*6 is the strict case.

A337600 is the unordered version.

A337603 does not consider a singleton to be coprime unless it is (1).

A337664 counts these compositions of any length.

A000740 counts relatively prime compositions.

A337561 counts pairwise coprime strict compositions.

A000217 counts 3-part compositions.

A001399/A069905/A211540 count 3-part partitions.

A023023 counts relatively prime 3-part partitions.

A051424 counts pairwise coprime or singleton partitions.

A101268 counts pairwise coprime or singleton compositions.

A305713 counts pairwise coprime strict partitions.

A327516 counts pairwise coprime partitions.

A333227 ranks pairwise coprime compositions.

A333228 ranks compositions whose distinct parts are pairwise coprime.

A337461 counts pairwise coprime 3-part compositions.

Cf. A000212, A007359, A087087, A284825, A302696, A304709, A304712, A307719, A328673, A335235, A335238, A337483, A337562, A337601.

Programs

Mathematica

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