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Number of unordered triples of distinct positive integers summing to n, any two of which have a common divisor > 1.
22

%I #11 Jan 13 2021 10:00:20

%S 0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,2,0,4,0,4,1,5,0,9,0,8,3,10,0,17,1,14,

%T 5,16,1,25,1,21,8,26,2,37,1,30,15,33,2,49,2,44,16,44,2,64,6,54,21,56,

%U 3,87,5,65,30,70,9,101,5,80,34,98,6,121,6,96,52

%N Number of unordered triples of distinct positive integers summing to n, any two of which have a common divisor > 1.

%H Fausto A. C. Cariboni, <a href="/A337605/b337605.txt">Table of n, a(n) for n = 0..10000</a>

%e The a(n) triples for n = 12, 16, 18, 22, 27, 55:

%e (6,4,2) (8,6,2) (8,6,4) (10,8,4) (12,9,6) (28,21,6)

%e (10,4,2) (9,6,3) (12,6,4) (15,9,3) (30,20,5)

%e (10,6,2) (12,8,2) (18,6,3) (35,15,5)

%e (12,4,2) (14,6,2) (40,10,5)

%e (16,4,2) (25,20,10)

%e (30,15,10)

%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}],UnsameQ@@#&&stabQ[#,CoprimeQ]&]],{n,0,100}]

%Y A014612 intersected with A318719 ranks these partitions.

%Y A220377 is the coprime instead of non-coprime version.

%Y A318717 counts these partitions of any length, ranked by A318719.

%Y A337599 is the non-strict version.

%Y A337604 is the ordered non-strict version.

%Y A337605*6 is the ordered version.

%Y A023023 counts relatively prime 3-part partitions

%Y A051424 counts pairwise coprime or singleton partitions.

%Y A200976 and A328673 count pairwise non-coprime partitions.

%Y A307719 counts pairwise coprime 3-part partitions.

%Y A327516 counts pairwise coprime partitions, with strict case A305713.

%Y Cf. A000217, A001399, A014612, A082024, A178472, A220377, A284825, A337461, A337561, A337667.

%K nonn

%O 0,17

%A _Gus Wiseman_, Sep 20 2020