%I #10 Feb 07 2021 10:48:59

%S 0,1,1,3,6,15,27,57,108,208,393,749,1415,2687,5076,9583,18088,34156,

%T 64511,121898,230368,435460,823376,1557420,2946931,5578109,10561987,

%U 20005126,37902509,71832372,136173266,258211602,489738622,929074445,1762899107,3345713031

%N Number of compositions of n whose distinct parts are pairwise coprime, 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="/A337665/b337665.txt">Table of n, a(n) for n = 0..220</a>

%e The a(1) = 1 through a(5) = 15 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) (1,2,2)

%e (1,3,1)

%e (2,1,2)

%e (2,2,1)

%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[Join@@Permutations/@Select[IntegerPartitions[n],CoprimeQ@@Union[#]&]],{n,0,15}]

%Y A000740 is a relatively prime instead of pairwise coprime version.

%Y A304709 is the unordered version.

%Y A333228 ranks these compositions.

%Y A337561 is the strict case.

%Y A337603 is the length-3 case.

%Y A337664 considers all singletons to be coprime.

%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 A337461 counts pairwise coprime length-3 compositions.

%Y Cf. A007359, A007360, A302569, A302696, A304712, A335235, A335238, A337562, A337600, A337601, A337602, A337695.

%K nonn

%O 0,4

%A _Gus Wiseman_, Sep 22 2020

%E a(26)-a(35) from _Alois P. Heinz_, Sep 29 2020