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Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.
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%I #14 Jan 21 2021 04:23:30

%S 0,0,0,1,3,6,10,9,18,16,24,21,43,24,51,31,54,42,94,45,102,55,99,69,

%T 163,66,150,88,168,96,265,93,228,121,246,126,337,132,315,169,342,162,

%U 487,165,420,217,411,213,619,207,558,259,540,258,784,264,654,325,660

%N Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

%H Fausto A. C. Cariboni, <a href="/A337602/b337602.txt">Table of n, a(n) for n = 0..10000</a>

%e The a(3) = 1 through a(8) = 18 triples:

%e (1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5) (1,1,6)

%e (1,2,1) (1,2,2) (1,2,3) (1,3,3) (1,2,5)

%e (2,1,1) (1,3,1) (1,3,2) (1,5,1) (1,3,4)

%e (2,1,2) (1,4,1) (2,2,3) (1,4,3)

%e (2,2,1) (2,1,3) (2,3,2) (1,5,2)

%e (3,1,1) (2,2,2) (3,1,3) (1,6,1)

%e (2,3,1) (3,2,2) (2,1,5)

%e (3,1,2) (3,3,1) (2,3,3)

%e (3,2,1) (5,1,1) (2,5,1)

%e (4,1,1) (3,1,4)

%e (3,2,3)

%e (3,3,2)

%e (3,4,1)

%e (4,1,3)

%e (4,3,1)

%e (5,1,2)

%e (5,2,1)

%e (6,1,1)

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],SameQ@@#||CoprimeQ@@Union[#]&]],{n,0,100}]

%Y The complement in A014311 of A337695 ranks these compositions.

%Y A220377*6 is the strict case.

%Y A337600 is the unordered version.

%Y A337603 does not consider a singleton to be coprime unless it is (1).

%Y A337664 counts these compositions of any length.

%Y A000740 counts relatively prime compositions.

%Y A337561 counts pairwise coprime strict compositions.

%Y A000217 counts 3-part compositions.

%Y A001399/A069905/A211540 count 3-part partitions.

%Y A023023 counts relatively prime 3-part partitions.

%Y A051424 counts pairwise coprime or singleton partitions.

%Y A101268 counts pairwise coprime or singleton compositions.

%Y A305713 counts pairwise coprime strict partitions.

%Y A327516 counts pairwise coprime partitions.

%Y A333227 ranks pairwise coprime compositions.

%Y A333228 ranks compositions whose distinct parts are pairwise coprime.

%Y A337461 counts pairwise coprime 3-part compositions.

%Y Cf. A000212, A007359, A087087, A284825, A302696, A304709, A304712, A307719, A328673, A335235, A335238, A337483, A337562, A337601.

%K nonn

%O 0,5

%A _Gus Wiseman_, Sep 20 2020