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%I #13 Jul 03 2020 06:32:00
%S 1,2,4,8,16,32,3,6,12,24,48,96,192,5,10,20,40,80,160,320,640,1280,
%T 2560,13,26,52,104,208,416,7,14,28,56,112,224,448,896,9,18,36,72,144,
%U 288,576,1152,2304,4608,9216,18432,36864,33,66,132,264,528,1056,2112
%N a(1) = 1; for n>1, a(n) = floor(a(n-1)^(1/3)) if that number is not already in the sequence, otherwise a(n) = 2*a(n-1).
%C If k is not in this sequence, then none of k^(3^t), k^(3^t)+1, ..., (k+1)^(3^t)-1 belong to the sequence. Because (k+1)^(3^k) > 2*k^(3^k), any m > k^(3^k) is not in the sequence, which is a contradiction to {a(n)} is not bounded above. Therefore, this sequence is a permutation of the natural numbers.
%H Rémy Sigrist, <a href="/A335836/b335836.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%t Nest[Append[#1, If[FreeQ[#1, #2], #2, 2 #1[[-1]] ]] & @@ {#, Floor[#[[-1]]^(1/3)]} &, {1}, 56] (* _Michael De Vlieger_, Jun 28 2020 *)
%o (PARI) lista(nn) = {my(k, v=vector(nn)); v[1]=1; for(n=2, nn, if(vecsearch(vecsort(v), k=sqrtnint(v[n-1], 3)), v[n]=2*v[n-1], v[n]=k)); v; }
%Y Cf. A048766, A114183, A213912.
%K nonn
%O 1,2
%A _Jinyuan Wang_, Jun 27 2020
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