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A318690 Matula-Goebel numbers of powerful uniform rooted trees.

%I A318690 #8 Sep 02 2018 08:21:10
%S A318690 1,2,3,4,5,7,8,9,11,16,17,19,23,25,27,31,32,36,49,53,59,64,67,81,83,
%T A318690 97,100,103,121,125,127,128,131,151,196,216,225,227,241,243,256,277,
%U A318690 289,311,331,343,361,419,431,441,484,509,512,529,541,563,625,661,691
%N A318690 Matula-Goebel numbers of powerful uniform rooted trees.
%C A318690 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 uniform rooted tree iff either n = 1 or n is a prime number whose prime index is a Matula-Goebel number of a powerful uniform rooted tree or n is a squarefree number taken to a power > 1 whose prime indices are all Matula-Goebel numbers of powerful uniform rooted trees.
%H A318690 Gus Wiseman, <a href="/A318690/a318690.png">The first 96 powerful uniform rooted trees.</a>
%e A318690 The sequence of all powerful uniform rooted trees together with their Matula-Goebel numbers begins:
%e A318690    1: o
%e A318690    2: (o)
%e A318690    3: ((o))
%e A318690    4: (oo)
%e A318690    5: (((o)))
%e A318690    7: ((oo))
%e A318690    8: (ooo)
%e A318690    9: ((o)(o))
%e A318690   11: ((((o))))
%e A318690   16: (oooo)
%e A318690   17: (((oo)))
%e A318690   19: ((ooo))
%e A318690   23: (((o)(o)))
%e A318690   25: (((o))((o)))
%e A318690   27: ((o)(o)(o))
%e A318690   31: (((((o)))))
%e A318690   32: (ooooo)
%e A318690   36: (oo(o)(o))
%e A318690   49: ((oo)(oo))
%t A318690 powunQ[n_]:=Or[n==1,If[PrimeQ[n],powunQ[PrimePi[n]],And[SameQ@@FactorInteger[n][[All,2]],Min@@FactorInteger[n][[All,2]]>1,And@@powunQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
%t A318690 Select[Range[100],powunQ]
%Y A318690 Cf. A000081, A001694, A061775, A072774, A214577, A317705, A317707, A317710, A317717, A317719, A318611, A318612, A318689, A318692.
%K A318690 nonn
%O A318690 1,2
%A A318690 _Gus Wiseman_, Aug 31 2018