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%I #11 Dec 25 2018 17:28:35
%S 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 96,99,108,124,127,135,148,157,160,162,173,176,192,193,197,200,223,
%U 229,242,244,248,250,251,275,279,283,284,288,296,297,320,333,352,353
%N Matula-Goebel numbers of aperiodic rooted trees with locally distinct multiplicities.
%C 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 Sequence of aperiodic rooted trees with locally distinct multiplicities preceded by their Matula-Goebel numbers begins:
%e 1: o
%e 2: (o)
%e 3: ((o))
%e 5: (((o)))
%e 11: ((((o))))
%e 12: (oo(o))
%e 18: (o(o)(o))
%e 20: (oo((o)))
%e 24: (ooo(o))
%e 31: (((((o)))))
%e 37: ((oo(o)))
%e 40: (ooo((o)))
%e 44: (oo(((o))))
%e 45: ((o)(o)((o)))
%e 48: (oooo(o))
%e 50: (o((o))((o)))
%t mgsbQ[n_]:=Or[n==1,And[UnsameQ@@Last/@FactorInteger[n],GCD@@Last/@FactorInteger[n]==1,And@@Cases[FactorInteger[n],{p_,_}:>mgsbQ[PrimePi[p]]]]];
%t Select[Range[100],mgsbQ]
%Y Cf. A000081, A004111, A007097, A007916, A061775, A276625, A301700 A303431, A316793, A316795, A316796.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jul 14 2018
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