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%I #12 Jan 19 2021 21:55:06
%S 0,0,0,0,0,0,1,0,3,1,6,0,13,0,15,7,21,0,37,0,39,16,45,0,73,6,66,28,81,
%T 0,130,6,105,46,120,21,181,6,153,67,189,12,262,6,213,118,231,12,337,
%U 21,306,121,303,12,433,57,369,154,378,18,583,30,435,217,465
%N Number of ordered triples of positive integers summing to n, any two of which have a common divisor > 1.
%C The first relatively prime triple (15,10,6) is counted under a(31).
%H Fausto A. C. Cariboni, <a href="/A337604/b337604.txt">Table of n, a(n) for n = 0..10000</a>
%e The a(6) = 1 through a(15) = 7 triples (empty columns indicated by dots, A = 10):
%e 222 . 224 333 226 . 228 . 22A 339
%e 242 244 246 248 366
%e 422 262 264 266 393
%e 424 282 284 555
%e 442 336 2A2 636
%e 622 363 428 663
%e 426 446 933
%e 444 464
%e 462 482
%e 624 626
%e 633 644
%e 642 662
%e 822 824
%e 842
%e A22
%t stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],stabQ[#,CoprimeQ]&]],{n,0,100}]
%Y A014311 intersected with A337666 ranks these compositions.
%Y A337667 counts these compositions of any length.
%Y A335402 lists the positions of zeros.
%Y A337461 is the coprime instead of non-coprime version.
%Y A337599 is the unordered version, with strict case A337605.
%Y A337605*6 is the strict version.
%Y A000741 counts relatively prime 3-part compositions.
%Y A101268 counts pairwise coprime or singleton compositions.
%Y A200976 and A328673 count pairwise non-relatively prime partitions.
%Y A307719 counts pairwise coprime 3-part partitions.
%Y A318717 counts pairwise non-coprime strict partitions.
%Y A333227 ranks pairwise coprime compositions.
%Y Cf. A000217, A001399, A014612, A051424, A082024, A178472, A220377, A284825, A305713, A327516, A333228, A337561.
%K nonn
%O 0,9
%A _Gus Wiseman_, Sep 20 2020
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