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A324841 Matula-Goebel numbers of fully recursively anti-transitive rooted trees.

%I A324841 #5 Mar 18 2019 08:16:02
%S A324841 1,2,3,4,5,7,8,9,11,16,17,19,21,23,25,27,31,32,35,49,51,53,57,59,63,
%T A324841 64,67,73,77,81,83,85,95,97,103,115,121,125,127,128,131,133,147,149,
%U A324841 153,159,161,171,175,177,187,189,201,209,217,227,233,241,243,245
%N A324841 Matula-Goebel numbers of fully recursively anti-transitive rooted trees.
%C A324841 An unlabeled rooted tree is fully recursively anti-transitive if no proper terminal subtree of any terminal subtree is a branch of the larger subtree.
%e A324841 The sequence of fully recursively anti-transitive rooted trees together with their Matula-Goebel numbers begins:
%e A324841    1: o
%e A324841    2: (o)
%e A324841    3: ((o))
%e A324841    4: (oo)
%e A324841    5: (((o)))
%e A324841    7: ((oo))
%e A324841    8: (ooo)
%e A324841    9: ((o)(o))
%e A324841   11: ((((o))))
%e A324841   16: (oooo)
%e A324841   17: (((oo)))
%e A324841   19: ((ooo))
%e A324841   21: ((o)(oo))
%e A324841   23: (((o)(o)))
%e A324841   25: (((o))((o)))
%e A324841   27: ((o)(o)(o))
%e A324841   31: (((((o)))))
%e A324841   32: (ooooo)
%e A324841   35: (((o))(oo))
%e A324841   49: ((oo)(oo))
%e A324841   51: ((o)((oo)))
%e A324841   53: ((oooo))
%e A324841   57: ((o)(ooo))
%e A324841   59: ((((oo))))
%e A324841   63: ((o)(o)(oo))
%t A324841 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A324841 fratQ[n_]:=And[Intersection[Union@@Rest[FixedPointList[Union@@primeMS/@#&,primeMS[n]]],primeMS[n]]=={},And@@fratQ/@primeMS[n]];
%t A324841 Select[Range[100],fratQ]
%Y A324841 Cf. A000081, A007097, A290689, A303431, A304360, A306844, A316502, A318185, A318186.
%Y A324841 Cf. A324695, A324751, A324756, A324758, A324766, A324768, A324769.
%Y A324841 Cf. A324838, A324840, A324844.
%K A324841 nonn
%O A324841 1,2
%A A324841 _Gus Wiseman_, Mar 17 2019