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

0,5

Links

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

Example

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)

Mathematica

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

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.

Sequence in context: A224523 A159785 A057338 * A077856 A239977 A074499

Adjacent sequences: A337600 A337601 A337602 * A337604 A337605 A337606

Keyword

nonn

Author

Gus Wiseman, Sep 20 2020

Status

approved