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A317705 Matula-Goebel numbers of series-reduced powerful rooted trees.

%I A317705 #20 Sep 10 2018 23:44:34
%S A317705 1,4,8,16,32,49,64,128,196,256,343,361,392,512,784,1024,1372,1444,
%T A317705 1568,2048,2401,2744,2809,2888,3136,4096,5488,5776,6272,6859,8192,
%U A317705 9604,10976,11236,11552,12544,16384,16807,17161,17689,19208,21952,22472,23104,25088
%N A317705 Matula-Goebel numbers of series-reduced powerful rooted trees.
%C A317705 A positive integer n is a Matula-Goebel number of a series-reduced powerful rooted tree iff either n = 1 or n is a powerful number (meaning its prime multiplicities are all greater than 1) whose prime indices are all Matula-Goebel numbers of series-reduced powerful rooted trees, where a prime index of n is a number m such that prime(m) divides n.
%H A317705 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Göbel numbers</a>
%e A317705 The sequence of Matula-Goebel numbers of series-reduced powerful rooted trees together with the corresponding trees begins:
%e A317705     1: o
%e A317705     4: (oo)
%e A317705     8: (ooo)
%e A317705    16: (oooo)
%e A317705    32: (ooooo)
%e A317705    49: ((oo)(oo))
%e A317705    64: (oooooo)
%e A317705   128: (ooooooo)
%e A317705   196: (oo(oo)(oo))
%e A317705   256: (oooooooo)
%e A317705   343: ((oo)(oo)(oo))
%e A317705   361: ((ooo)(ooo))
%e A317705   392: (ooo(oo)(oo))
%e A317705   512: (ooooooooo)
%e A317705   784: (oooo(oo)(oo))
%t A317705 powgoQ[n_]:=Or[n==1,And[Min@@FactorInteger[n][[All,2]]>1,And@@powgoQ/@PrimePi/@FactorInteger[n][[All,1]]]];
%t A317705 Select[Range[1000],powgoQ] (* _Gus Wiseman_, Aug 31 2018 *)
%t A317705 (* Second program: *)
%t A317705 Nest[Function[a, Append[a, Block[{k = a[[-1]] + 1}, While[Nand[AllTrue[#[[All, -1]], # > 1 & ], AllTrue[PrimePi[#[[All, 1]] ], MemberQ[a, #] &]] &@ FactorInteger@ k, k++]; k]]], {1}, 44] (* _Michael De Vlieger_, Aug 05 2018 *)
%Y A317705 Cf. A000081, A001694, A061775, A111299, A214577, A276625, A277098, A303431.
%Y A317705 Cf. A317102, A317707, A317708, A317709, A317710, A317711, A317712, A317717, A317718, A317719.
%K A317705 nonn
%O A317705 1,2
%A A317705 _Gus Wiseman_, Aug 04 2018
%E A317705 Rewritten by _Gus Wiseman_, Aug 31 2018