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%I #13 Jan 12 2021 05:53:02
%S 0,0,0,0,0,0,0,0,0,0,1,0,2,0,2,1,4,0,7,1,7,3,9,2,15,3,13,5,17,4,29,5,
%T 20,8,28,8,42,8,31,14,42,10,59,12,45,21,52,14,77,17,68,26,69,19,101,
%U 26,84,34,86,25,138,28,95,43,111,36,161,35,118,52,151
%N Number of pairwise coprime unordered triples of positive integers > 1 summing to n.
%C Such partitions are necessarily strict.
%C The Heinz numbers of these partitions are the intersection of A005408 (no 1's), A014612 (triples), and A302696 (coprime).
%H Fausto A. C. Cariboni, <a href="/A337563/b337563.txt">Table of n, a(n) for n = 0..10000</a>
%e The a(10) = 1 through a(24) = 15 triples (empty columns indicated by dots, A..J = 10..19):
%e 532 . 543 . 743 753 754 . 765 B53 875 975 985 B75 987
%e 732 752 853 873 974 B73 B65 D73 B76
%e 952 954 A73 D53 B74 B85
%e B32 972 B54 B83 B94
%e B43 B72 B92 BA3
%e B52 D43 D54 C75
%e D32 D52 D72 D65
%e E53 D74
%e H32 D83
%e D92
%e F72
%e G53
%e H43
%e H52
%e J32
%t Table[Length[Select[IntegerPartitions[n,{3}],!MemberQ[#,1]&&CoprimeQ@@#&]],{n,0,30}]
%Y A055684 is the version for pairs.
%Y A220377 allows 1's, with non-strict version A307719.
%Y A337485 counts these partitions of any length.
%Y A337563*6 is the ordered version.
%Y A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
%Y A002865 counts partitions with no 1's, with strict case A025147.
%Y A007359 counts pairwise coprime partitions with no 1's.
%Y A078374 counts relatively prime strict partitions.
%Y A200976 and A328673 count pairwise non-coprime partitions.
%Y A302696 ranks pairwise coprime partitions.
%Y A302698 counts relatively prime partitions with no 1's.
%Y A305713 counts pairwise coprime strict partitions.
%Y A327516 counts pairwise coprime partitions.
%Y A337452 counts relatively prime strict partitions with no 1's.
%Y Cf. A007304, A082024, A101268, A284825, A332004, A337451, A337461, A337462, A337561, A337599, A337601, A337605.
%K nonn
%O 0,13
%A _Gus Wiseman_, Sep 21 2020
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