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A316794 Matula-Goebel numbers of aperiodic rooted trees with locally distinct multiplicities.

%I A316794 #11 Dec 25 2018 17:28:35
%S A316794 1,2,3,5,11,12,18,20,24,31,37,40,44,45,48,50,54,61,71,72,75,80,88,89,
%T A316794 96,99,108,124,127,135,148,157,160,162,173,176,192,193,197,200,223,
%U A316794 229,242,244,248,250,251,275,279,283,284,288,296,297,320,333,352,353
%N A316794 Matula-Goebel numbers of aperiodic rooted trees with locally distinct multiplicities.
%C A316794 A positive integer belongs to the sequence iff either it is equal to 1 or it belongs to A007916 (numbers that are not perfect powers, or numbers whose prime multiplicities are relatively prime) as well as to A130091 (numbers whose prime multiplicities are distinct), and all of its prime indices already belong to the sequence. A prime index of n is a number m such that prime(m) divides n.
%e A316794 Sequence of aperiodic rooted trees with locally distinct multiplicities preceded by their Matula-Goebel numbers begins:
%e A316794    1: o
%e A316794    2: (o)
%e A316794    3: ((o))
%e A316794    5: (((o)))
%e A316794   11: ((((o))))
%e A316794   12: (oo(o))
%e A316794   18: (o(o)(o))
%e A316794   20: (oo((o)))
%e A316794   24: (ooo(o))
%e A316794   31: (((((o)))))
%e A316794   37: ((oo(o)))
%e A316794   40: (ooo((o)))
%e A316794   44: (oo(((o))))
%e A316794   45: ((o)(o)((o)))
%e A316794   48: (oooo(o))
%e A316794   50: (o((o))((o)))
%t A316794 mgsbQ[n_]:=Or[n==1,And[UnsameQ@@Last/@FactorInteger[n],GCD@@Last/@FactorInteger[n]==1,And@@Cases[FactorInteger[n],{p_,_}:>mgsbQ[PrimePi[p]]]]];
%t A316794 Select[Range[100],mgsbQ]
%Y A316794 Cf. A000081, A004111, A007097, A007916, A061775, A276625, A301700  A303431, A316793, A316795, A316796.
%K A316794 nonn
%O A316794 1,2
%A A316794 _Gus Wiseman_, Jul 14 2018