A335402 -id:A335402 - cp's OEIS frontend

A337604 Number of ordered triples of positive integers summing to n, any two of which have a common divisor > 1.

0, 0, 0, 0, 0, 0, 1, 0, 3, 1, 6, 0, 13, 0, 15, 7, 21, 0, 37, 0, 39, 16, 45, 0, 73, 6, 66, 28, 81, 0, 130, 6, 105, 46, 120, 21, 181, 6, 153, 67, 189, 12, 262, 6, 213, 118, 231, 12, 337, 21, 306, 121, 303, 12, 433, 57, 369, 154, 378, 18, 583, 30, 435, 217, 465

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

The first relatively prime triple (15,10,6) is counted under a(31).

			The a(6) = 1 through a(15) = 7 triples (empty columns indicated by dots, A = 10):
  222  .  224  333  226  .  228  .  22A  339
          242       244     246     248  366
          422       262     264     266  393
                    424     282     284  555
                    442     336     2A2  636
                    622     363     428  663
                            426     446  933
                            444     464
                            462     482
                            624     626
                            633     644
                            642     662
                            822     824
                                    842
                                    A22

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

A014311 intersected with A337666 ranks these compositions.
A337667 counts these compositions of any length.
A335402 lists the positions of zeros.
A337461 is the coprime instead of non-coprime version.
A337599 is the unordered version, with strict case A337605.
A337605*6 is the strict version.
A000741 counts relatively prime 3-part compositions.
A101268 counts pairwise coprime or singleton compositions.
A200976 and A328673 count pairwise non-relatively prime partitions.
A307719 counts pairwise coprime 3-part partitions.
A318717 counts pairwise non-coprime strict partitions.
A333227 ranks pairwise coprime compositions.
Cf. A000217, A001399, A014612, A051424, A082024, A178472, A220377, A284825, A305713, A327516, A333228, A337561.

stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],stabQ[#,CoprimeQ]&]],{n,0,100}]

A337599 Number of unordered triples of positive integers summing to n, any two of which have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 4, 0, 4, 3, 5, 0, 9, 0, 9, 5, 10, 0, 16, 2, 14, 7, 17, 0, 27, 1, 21, 11, 24, 6, 36, 1, 30, 15, 37, 2, 51, 1, 41, 25, 44, 2, 64, 5, 58, 25, 57, 2, 81, 13, 69, 31, 70, 3, 108, 5, 80, 43, 85, 17, 123, 5, 97, 46, 120, 6, 144, 6

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

First differs from A082024 at a(31) = 1, A082024(31) = 0.

The first relatively prime triple is (15,10,6), counted under a(31).

			The a(6) = 1 through a(16) = 5 partitions are (empty columns indicated by dots, A..G = 10..16):
  222  .  422  333  442  .  444  .  644  555  664  .  666  .  866
                    622     633     662  663  844     864     884
                            642     842  933  862     882     A55
                            822     A22       A42     963     A64
                                              C22     A44     A82
                                                      A62     C44
                                                      C33     C62
                                                      C42     E42
                                                      E22     G22

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

A014612 intersected with A337694 ranks these partitions.
A200976 and A328673 count these partitions of any length.
A284825 is the case that is also relatively prime.
A307719 is the pairwise coprime instead of non-coprime version.
A335402 gives the positions of zeros.
A337604 is the ordered version.
A337605 is the strict case.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts strict pairwise coprime partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000212, A000217, A001840, A018783, A082024, A211540, A220377, A337461.

stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
Table[Length[Select[IntegerPartitions[n,{3}],stabQ[#,CoprimeQ]&]],{n,0,100}]

cp's OEIS Frontend

A337604 Number of ordered triples of positive integers summing to n, any two of which have a common divisor > 1.

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