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%I #12 Jan 13 2021 14:28:44
%S 0,0,0,0,0,0,1,0,1,1,2,0,4,0,4,3,5,0,9,0,9,5,10,0,16,2,14,7,17,0,27,1,
%T 21,11,24,6,36,1,30,15,37,2,51,1,41,25,44,2,64,5,58,25,57,2,81,13,69,
%U 31,70,3,108,5,80,43,85,17,123,5,97,46,120,6,144,6
%N Number of unordered triples of positive integers summing to n, any two of which have a common divisor > 1.
%C First differs from A082024 at a(31) = 1, A082024(31) = 0.
%C The first relatively prime triple is (15,10,6), counted under a(31).
%H Fausto A. C. Cariboni, <a href="/A337599/b337599.txt">Table of n, a(n) for n = 0..10000</a>
%e The a(6) = 1 through a(16) = 5 partitions are (empty columns indicated by dots, A..G = 10..16):
%e 222 . 422 333 442 . 444 . 644 555 664 . 666 . 866
%e 622 633 662 663 844 864 884
%e 642 842 933 862 882 A55
%e 822 A22 A42 963 A64
%e C22 A44 A82
%e A62 C44
%e C33 C62
%e C42 E42
%e E22 G22
%t stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
%t Table[Length[Select[IntegerPartitions[n,{3}],stabQ[#,CoprimeQ]&]],{n,0,100}]
%Y A014612 intersected with A337694 ranks these partitions.
%Y A200976 and A328673 count these partitions of any length.
%Y A284825 is the case that is also relatively prime.
%Y A307719 is the pairwise coprime instead of non-coprime version.
%Y A335402 gives the positions of zeros.
%Y A337604 is the ordered version.
%Y A337605 is the strict case.
%Y A051424 counts pairwise coprime or singleton partitions.
%Y A101268 counts pairwise coprime or singleton compositions.
%Y A305713 counts strict pairwise coprime partitions.
%Y A327516 counts pairwise coprime partitions.
%Y A333227 ranks pairwise coprime compositions.
%Y A333228 ranks compositions whose distinct parts are pairwise coprime.
%Y Cf. A000212, A000217, A001840, A018783, A082024, A211540, A220377, A337461.
%K nonn
%O 0,11
%A _Gus Wiseman_, Sep 20 2020
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