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%I #5 Mar 18 2019 08:15:16
%S 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 40,44,46,49,50,51,53,57,59,62,63,64,67,68,71,73,77,79,80,81,83,85,87,
%U 88,92,93,95,97,99,100,103,109,115,118,121,124,125,127,128
%N Matula-Goebel numbers of recursively anti-transitive rooted trees.
%C The complement is {6, 12, 13, 14, 15, 18, 24, 26, 28, 30, 36, ...}.
%C 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 The sequence of recursively 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 10: (o((o)))
%e 11: ((((o))))
%e 16: (oooo)
%e 17: (((oo)))
%e 19: ((ooo))
%e 20: (oo((o)))
%e 21: ((o)(oo))
%e 22: (o(((o))))
%e 23: (((o)(o)))
%e 25: (((o))((o)))
%e 27: ((o)(o)(o))
%e 29: ((o((o))))
%e 31: (((((o)))))
%e 32: (ooooo)
%e 33: ((o)(((o))))
%e 34: (o((oo)))
%e 35: (((o))(oo))
%e 40: (ooo((o)))
%e 44: (oo(((o))))
%e 46: (o((o)(o)))
%e 49: ((oo)(oo))
%e 50: (o((o))((o)))
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t totantiQ[n_]:=And[Intersection[Union@@primeMS/@primeMS[n],primeMS[n]]=={},And@@totantiQ/@primeMS[n]];
%t Select[Range[100],totantiQ]
%Y Cf. A007097, A000081, A290689, A303431, A304360, A306844, A316502, A318186.
%Y Cf. A324695, A324751, A324756, A324758, A324765, A324767, A324769, A324838, A324841, A324844.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 17 2019
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