%I #8 Aug 05 2018 08:24:41
%S 1,1,1,2,4,10,20,48,108,255,595,1435,3434,8372,20419,50289,124289,
%T 309122,771508,1934462
%N Number of aperiodic relatively prime trees with n nodes.
%C An unlabeled rooted tree is aperiodic and relatively prime iff either it is a single node or a single node with a single aperiodic relatively prime branch, or the branches directly under any given node have empty intersection (relatively prime) and also have relatively prime multiplicities (aperiodic) and are themselves aperiodic relatively prime trees.
%H Gus Wiseman, <a href="/A317708/a317708.png">All 48 aperiodic relatively prime trees with 8 nodes.</a>
%e The a(6) = 10 aperiodic relatively prime trees:
%e (((((o)))))
%e (((o(o))))
%e ((o((o))))
%e ((oo(o)))
%e (o(((o))))
%e (o(o(o)))
%e ((o)((o)))
%e (oo((o)))
%e (o(o)(o))
%e (ooo(o))
%t rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,And[Intersection@@#=={},GCD@@Length/@Split[#]==1]]&],{ptn,IntegerPartitions[n-1]}]];
%t Table[Length[rurt[n]],{n,10}]
%Y Cf. A000081, A001190, A004111, A301700, A303431, A316501, A316500.
%Y Cf. A317705, A317707, A317709, A317710, A317711, A317712.
%K nonn,more
%O 1,4
%A _Gus Wiseman_, Aug 05 2018