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Numbers that are not powerful tree numbers.
4

%I #9 Sep 10 2018 23:44:48

%S 6,10,12,13,14,15,18,20,21,22,24,26,28,29,30,33,34,35,37,38,39,40,41,

%T 42,43,44,45,46,47,48,50,51,52,54,55,56,57,58,60,61,62,63,65,66,68,69,

%U 70,71,73,74,75,76,77,78,79,80,82,84,85,86,87,88,89,90,91

%N Numbers that are not powerful tree numbers.

%C A positive integer n is a powerful tree number iff either n = 1 or n is a prime number whose prime index is a powerful tree number, or n is a powerful number (meaning its prime multiplicities are all greater than 1) whose prime indices are all powerful tree numbers. A prime index of n is a number m such that prime(m) divides n.

%e The sequence of numbers that are not powerful tree numbers together with their Matula-Goebel trees begins:

%e 6: (o(o))

%e 10: (o((o)))

%e 12: (oo(o))

%e 13: ((o(o)))

%e 14: (o(oo))

%e 15: ((o)((o)))

%e 18: (o(o)(o))

%e 20: (oo((o)))

%e 21: ((o)(oo))

%e 22: (o(((o))))

%e 24: (ooo(o))

%e 26: (o(o(o)))

%e 28: (oo(oo))

%e 29: ((o((o))))

%e 30: (o(o)((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[100],!powgoQ[#]&]

%Y Complement of A318612.

%Y Cf. A000081, A001694, A061775, A111299, A214577, A276625, A277098, A303431.

%Y Cf. A317102, A317707, A317708, A317709, A317710, A317711, A317712, A317717, A317718.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 05 2018