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%I #9 May 05 2018 02:24:49
%S 1,2,3,5,6,10,11,12,13,15,18,20,22,24,26,29,30,31,33,37,39,40,41,44,
%T 45,47,48,50,52,54,55,58,60,61,62,65,66,71,72,74,75,78,79,80,82,87,88,
%U 89,90,93,94,96,99,101,104,108,109,110,111,113,116,117,120,122
%N Aperiodic tree numbers. Matula-Goebel numbers of aperiodic rooted trees.
%C A positive integer is an aperiodic tree number 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) and all of its prime indices are also aperiodic tree numbers, where a prime index of n is a number m such that prime(m) divides n.
%e Sequence of aperiodic rooted trees begins:
%e 01 o
%e 02 (o)
%e 03 ((o))
%e 05 (((o)))
%e 06 (o(o))
%e 10 (o((o)))
%e 11 ((((o))))
%e 12 (oo(o))
%e 13 ((o(o)))
%e 15 ((o)((o)))
%e 18 (o(o)(o))
%e 20 (oo((o)))
%e 22 (o(((o))))
%e 24 (ooo(o))
%e 26 (o(o(o)))
%e 29 ((o((o))))
%e 30 (o(o)((o)))
%e 31 (((((o)))))
%e 33 ((o)(((o))))
%t zapQ[1]:=True;zapQ[n_]:=And[GCD@@FactorInteger[n][[All,2]]===1,And@@zapQ/@PrimePi/@FactorInteger[n][[All,1]]];
%t Select[Range[100],zapQ]
%Y Cf. A000081, A000740, A000837, A007097, A007916, A052409, A052410, A061775, A111299, A214577, A275024, A276625, A290760, A290822, A291442, A298533, A298536, A301700, A302242, A303386.
%K nonn
%O 1,2
%A _Gus Wiseman_, Apr 23 2018
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