login
Number of pairwise coprime compositions of n, where a singleton is not considered coprime unless it is (1).
26

%I #10 Feb 03 2021 23:10:29

%S 1,1,1,3,6,12,21,37,62,100,159,253,402,634,983,1491,2224,3280,4813,

%T 7043,10270,14888,21415,30585,43400,61204,85747,119295,164834,226422,

%U 309663,422301,574826,781236,1060181,1436367,1942588,2622078,3531151,4742315,6348410

%N Number of pairwise coprime compositions of n, where a singleton is not considered coprime unless it is (1).

%C A composition of n is a finite sequence of positive integers summing to n.

%H Fausto A. C. Cariboni, <a href="/A337462/b337462.txt">Table of n, a(n) for n = 0..500</a>

%F For n > 1, a(n) = A101268(n) - 1.

%e The a(1) = 1 through a(5) = 12 compositions:

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

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

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

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

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

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

%e (3,1,1)

%e (1,1,1,2)

%e (1,1,2,1)

%e (1,2,1,1)

%e (2,1,1,1)

%e (1,1,1,1,1)

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],#=={}||CoprimeQ@@#&]],{n,0,10}]

%Y A000740 counts the relatively prime instead of pairwise coprime version.

%Y A101268 considers all singletons to be coprime, with strict case A337562.

%Y A327516 is the unordered version.

%Y A333227 ranks these compositions, with complement A335239.

%Y A337461 counts these compositions of length 3.

%Y A337561 is the strict case.

%Y A051424 counts pairwise coprime or singleton partitions.

%Y A101268 counts pairwise coprime or singleton compositions.

%Y A178472 counts compositions with a common factor.

%Y A305713 counts strict pairwise coprime partitions.

%Y A328673 counts pairwise non-coprime partitions.

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

%Y A337667 counts pairwise non-coprime compositions.

%Y Cf. A001523, A007360, A087087, A220377, A302569, A307719, A326675, A335235, A335238, A337664.

%K nonn

%O 0,4

%A _Gus Wiseman_, Sep 18 2020