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A324766 Matula-Goebel numbers of recursively anti-transitive rooted trees.

%I A324766 #5 Mar 18 2019 08:15:16
%S A324766 1,2,3,4,5,7,8,9,10,11,16,17,19,20,21,22,23,25,27,29,31,32,33,34,35,
%T A324766 40,44,46,49,50,51,53,57,59,62,63,64,67,68,71,73,77,79,80,81,83,85,87,
%U A324766 88,92,93,95,97,99,100,103,109,115,118,121,124,125,127,128
%N A324766 Matula-Goebel numbers of recursively anti-transitive rooted trees.
%C A324766 The complement is {6, 12, 13, 14, 15, 18, 24, 26, 28, 30, 36, ...}.
%C A324766 An unlabeled rooted tree is recursively anti-transitive if no branch of a branch of a terminal subtree is a branch of the same subtree.
%e A324766 The sequence of recursively anti-transitive rooted trees together with their Matula-Goebel numbers begins:
%e A324766    1: o
%e A324766    2: (o)
%e A324766    3: ((o))
%e A324766    4: (oo)
%e A324766    5: (((o)))
%e A324766    7: ((oo))
%e A324766    8: (ooo)
%e A324766    9: ((o)(o))
%e A324766   10: (o((o)))
%e A324766   11: ((((o))))
%e A324766   16: (oooo)
%e A324766   17: (((oo)))
%e A324766   19: ((ooo))
%e A324766   20: (oo((o)))
%e A324766   21: ((o)(oo))
%e A324766   22: (o(((o))))
%e A324766   23: (((o)(o)))
%e A324766   25: (((o))((o)))
%e A324766   27: ((o)(o)(o))
%e A324766   29: ((o((o))))
%e A324766   31: (((((o)))))
%e A324766   32: (ooooo)
%e A324766   33: ((o)(((o))))
%e A324766   34: (o((oo)))
%e A324766   35: (((o))(oo))
%e A324766   40: (ooo((o)))
%e A324766   44: (oo(((o))))
%e A324766   46: (o((o)(o)))
%e A324766   49: ((oo)(oo))
%e A324766   50: (o((o))((o)))
%t A324766 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A324766 totantiQ[n_]:=And[Intersection[Union@@primeMS/@primeMS[n],primeMS[n]]=={},And@@totantiQ/@primeMS[n]];
%t A324766 Select[Range[100],totantiQ]
%Y A324766 Cf. A007097, A000081, A290689, A303431, A304360, A306844, A316502, A318186.
%Y A324766 Cf. A324695, A324751, A324756, A324758, A324765, A324767, A324769, A324838, A324841, A324844.
%K A324766 nonn
%O A324766 1,2
%A A324766 _Gus Wiseman_, Mar 17 2019