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Number of compositions of n where any two parts have a common divisor > 1.
16

%I #13 Feb 09 2021 18:38:31

%S 1,0,1,1,2,1,5,1,8,4,17,1,38,1,65,19,128,1,284,1,518,67,1025,1,2168,

%T 16,4097,256,8198,1,16907,7,32768,1027,65537,79,133088,19,262145,4099,

%U 524408,25,1056731,51,2097158,16636,4194317,79,8421248,196,16777712

%N Number of compositions of n where any two parts have a common divisor > 1.

%C First differs from A178472 at a(31) = 7, a(31) = 1.

%H Fausto A. C. Cariboni, <a href="/A337667/b337667.txt">Table of n, a(n) for n = 0..290</a>

%e The a(2) = 1 through a(10) = 17 compositions (A = 10):

%e 2 3 4 5 6 7 8 9 A

%e 22 24 26 36 28

%e 33 44 63 46

%e 42 62 333 55

%e 222 224 64

%e 242 82

%e 422 226

%e 2222 244

%e 262

%e 424

%e 442

%e 622

%e 2224

%e 2242

%e 2422

%e 4222

%e 22222

%t stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];

%t Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],stabQ[#,CoprimeQ]&]],{n,0,15}]

%Y A101268 = 1 + A337462 is the pairwise coprime version.

%Y A328673 = A200976 + 1 is the unordered version.

%Y A337604 counts these compositions of length 3.

%Y A337666 ranks these compositions.

%Y A337694 gives Heinz numbers of the unordered version.

%Y A337983 is the strict case.

%Y A051185 counts intersecting set-systems, with spanning case A305843.

%Y A318717 is the unordered strict case.

%Y A319786 is the version for factorizations, with strict case A318749.

%Y A327516 counts pairwise coprime partitions.

%Y A333227 ranks pairwise coprime compositions.

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

%Y Cf. A051424, A082024, A178472, A284825, A319752, A327039, A337599, A337605, A337665.

%K nonn

%O 0,5

%A _Gus Wiseman_, Oct 05 2020