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A261205 Numbers k such that floor(k^(1/m)) divides k for all integers m >= 1.

%I A261205 #48 Jul 16 2021 03:29:56
%S A261205 1,2,3,4,6,8,12,16,20,24,30,36,42,48,64,72,80,120,210,240,288,324,420,
%T A261205 528,552,576,600,624,900,1260,1764,1848,1980,3024,6480,8100,8280,
%U A261205 11880,14160,14280,14400,14520,14640,28560,43680,44520,46872,50400,175560,331200,346920,491400,809100,3418800,4772040,38937600,203918400,2000862360
%N A261205 Numbers k such that floor(k^(1/m)) divides k for all integers m >= 1.
%C A261205 Is this a finite sequence?
%C A261205 There are no other terms below 10^23. - _Giovanni Resta_, Aug 13 2015
%e A261205 From _Michel Marcus_, Aug 13 2015: (Start)
%e A261205 For k=1 to 9, we have the following floored roots:
%e A261205   k=1: 1, 1, ...
%e A261205   k=2: 2, 1, 1, ...
%e A261205   k=3: 3, 1, 1, ...
%e A261205   k=4: 4, 2, 1, 1, ...
%e A261205   k=5: 5, 2, 1, 1, ...
%e A261205   k=6: 6, 2, 1, 1, ...
%e A261205   k=7: 7, 2, 1, 1, ...
%e A261205   k=8: 8, 2, 2, 1, 1, ...
%e A261205   k=9: 9, 3, 2, 1, 1, ...
%e A261205 where one can see that 5, 7 and 9 are not terms. (End)
%t A261205 fQ[n_] := Block[{d, k = 2, lst = {}}, While[d = Floor[n^(1/k)]; d > 1, AppendTo[lst, d]; k++]; Union[ IntegerQ@# & /@ (n/Union[lst])] == {True}]; k = 4; lst = {1, 2, 3}; While[k < 10^6, If[fQ@ k, AppendTo[lst, k]; Print@ k]; k++]; lst (* _Robert G. Wilson v_, Aug 15 2015 *)
%o A261205 (PARI) is(n) = my(k,t); k=2; while( (t=sqrtnint(n, k)) > 1, if(n%t, return(0)); k++); 1
%o A261205 n=1; while(n<10^5,if(is(n),print1(n,", "));n++) /* Able to generate terms < 10^5 */ \\ _Derek Orr_, Aug 12 2015
%Y A261205 Cf. A261206, A261341, A261342.
%Y A261205 Subsequence of A006446.
%K A261205 nonn,nice
%O A261205 1,2
%A A261205 Yan A. Denenberg and _Max Alekseyev_, Aug 11 2015