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%I #4 Sep 02 2018 08:20:54
%S 1,2,3,4,5,7,8,9,11,16,17,19,23,25,27,31,32,36,49,53,59,64,67,72,81,
%T 83,97,100,103,108,121,125,127,128,131,144,151,196,200,216,225,227,
%U 241,243,256,277,288,289,311,324,331,343,359,361,392,400,419,431,432
%N Matula-Goebel numbers of powerful rooted trees.
%C A prime index of n is a number m such that prime(m) divides n. A positive integer n is a Matula-Goebel number of a powerful rooted tree iff either n = 1 or n is a prime number whose prime index is a Matula-Goebel number of a powerful rooted tree or n is a powerful number (meaning its prime multiplicities are all greater than 1) whose prime indices are all Matula-Goebel numbers of powerful rooted trees.
%e The sequence of all powerful 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 11: ((((o))))
%e 16: (oooo)
%e 17: (((oo)))
%e 19: ((ooo))
%e 23: (((o)(o)))
%e 25: (((o))((o)))
%e 27: ((o)(o)(o))
%e 31: (((((o)))))
%t powgoQ[n_]:=Or[n==1,If[PrimeQ[n],powgoQ[PrimePi[n]],And[Min@@FactorInteger[n][[All,2]]>1,And@@powgoQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
%t Select[Range[1000],powgoQ]
%Y Complement of A317719.
%Y Cf. A000081, A001694, A061775, A111299, A214577, A276625, A303431, A317102, A317705, A317707.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 30 2018
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