%I #12 Jan 21 2021 04:23:39
%S 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,
%T 66,150,87,168,96,264,93,228,120,246,126,336,132,315,168,342,162,486,
%U 165,420,216,411,213,618,207,558,258,540,258,783,264,654,324,660
%N 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).
%H Fausto A. C. Cariboni, <a href="/A337603/b337603.txt">Table of n, a(n) for n = 0..10000</a>
%e The a(3) = 1 through a(8) = 18 triples:
%e (1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5) (1,1,6)
%e (1,2,1) (1,2,2) (1,2,3) (1,3,3) (1,2,5)
%e (2,1,1) (1,3,1) (1,3,2) (1,5,1) (1,3,4)
%e (2,1,2) (1,4,1) (2,2,3) (1,4,3)
%e (2,2,1) (2,1,3) (2,3,2) (1,5,2)
%e (3,1,1) (2,3,1) (3,1,3) (1,6,1)
%e (3,1,2) (3,2,2) (2,1,5)
%e (3,2,1) (3,3,1) (2,3,3)
%e (4,1,1) (5,1,1) (2,5,1)
%e (3,1,4)
%e (3,2,3)
%e (3,3,2)
%e (3,4,1)
%e (4,1,3)
%e (4,3,1)
%e (5,1,2)
%e (5,2,1)
%e (6,1,1)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],CoprimeQ@@Union[#]&]],{n,0,100}]
%Y A014311 intersected with A333228 ranks these compositions.
%Y A220377*6 is the strict case.
%Y A337461 is the strict case except for any number of 1's.
%Y A337601 is the unordered version.
%Y A337602 considers all singletons to be coprime.
%Y A337665 counts these compositions of any length, ranked by A333228 with complement A335238.
%Y A000217(n - 2) counts 3-part compositions.
%Y A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
%Y A007318 and A097805 count compositions by length.
%Y A051424 counts pairwise coprime or singleton partitions.
%Y A101268 counts pairwise coprime or singleton compositions.
%Y A304711 ranks partitions whose distinct parts are pairwise coprime.
%Y A305713 counts strict pairwise coprime partitions.
%Y A327516 counts pairwise coprime partitions, with strict case A305713.
%Y A333227 ranks pairwise coprime compositions.
%Y Cf. A000740, A001840, A007359, A087087, A178472, A284825, A302696, A307719, A335235, A337561, A337695.
%K nonn
%O 0,5
%A _Gus Wiseman_, Sep 20 2020