%I #4 Mar 18 2019 08:15:37

%S 1,2,3,4,5,7,8,9,11,13,16,17,19,21,23,25,27,29,31,32,35,37,41,43,47,

%T 49,51,53,57,59,61,63,64,65,67,71,73,77,79,81,83,85,89,91,95,97,101,

%U 103,107,109,113,115,121,125,127,128,129,131,133,137,139,143,147

%N Matula-Goebel numbers of fully anti-transitive rooted trees.

%C An unlabeled rooted tree is fully anti-transitive if no proper terminal subtree of any branch of the root is a branch of the root.

%e The sequence of fully anti-transitive rooted trees together with their Matula-Goebel numbers begins:

%e 1: o

%e 2: (o)

%e 3: ((o))

%e 4: (oo)

%e 5: (((o)))

%e 7: ((oo))

%e 8: (ooo)

%e 9: ((o)(o))

%e 11: ((((o))))

%e 13: ((o(o)))

%e 16: (oooo)

%e 17: (((oo)))

%e 19: ((ooo))

%e 21: ((o)(oo))

%e 23: (((o)(o)))

%e 25: (((o))((o)))

%e 27: ((o)(o)(o))

%e 29: ((o((o))))

%e 31: (((((o)))))

%e 32: (ooooo)

%e 35: (((o))(oo))

%e 37: ((oo(o)))

%e 41: (((o(o))))

%e 43: ((o(oo)))

%e 47: (((o)((o))))

%e 49: ((oo)(oo))

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t fullantiQ[n_]:=Intersection[Union@@Rest[FixedPointList[Union@@primeMS/@#&,primeMS[n]]],primeMS[n]]=={};

%t Select[Range[100],fullantiQ]

%Y Cf. A000081, A007097, A276625, A290760, A304360, A306844.

%Y Cf. A324695, A324751, A324756, A324758, A324766, A324768, A324770.

%Y Cf. A324838, A324841, A324845, A324846.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 17 2019