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A337563 Number of pairwise coprime unordered triples of positive integers > 1 summing to n.

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