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%I #8 Sep 02 2018 08:21:10
%S 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 97,100,103,121,125,127,128,131,151,196,216,225,227,241,243,256,277,
%U 289,311,331,343,361,419,431,441,484,509,512,529,541,563,625,661,691
%N Matula-Goebel numbers of powerful uniform 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 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 Gus Wiseman, <a href="/A318690/a318690.png">The first 96 powerful uniform rooted trees.</a>
%e The sequence of all powerful uniform 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)))))
%e 32: (ooooo)
%e 36: (oo(o)(o))
%e 49: ((oo)(oo))
%t 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 Select[Range[100],powunQ]
%Y Cf. A000081, A001694, A061775, A072774, A214577, A317705, A317707, A317710, A317717, A317719, A318611, A318612, A318689, A318692.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 31 2018
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