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Number of pairwise coprime unordered triples of positive integers > 1 summing to n.
21

%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