Permutation? Floor Of Ratio Of Adjacent Terms = 2

[Maximilian Hasler]

>  Is there always an unused positive integer, a(n),
>  such that
>  floor(max(a(n),a(n-1))/min(a(n),a(n-1))) = 2, or
>  does the sequence terminate? If the sequence is
>  infinite, is it a permutation of the positive
>  integers?

as complement to my reply, note that the smallest numbers not used at
the 50000-th iteration are:
for(j=1,29999,bittest(t,j)|print1(j","))
29489,29492,29496,29497,29508,29513,29521,29524,29527,29529,29532,29536,29537,29540,29545,29568,29571,29573,29576,29577,
29580,29587,29592,29593,29600,29601,29603,29608,29609,29612,29619,29633,29635,29639,29640,29641,29643,29644,29659,29660,
29663,29665,29671,29673,29676,29677,29679,29681,29684,29691,29692,29699,29700,29701,29705,29707,29708,29713,29715,29716,
29720,29721,29737,29740,29743,29745,29747,29748,29749,29753,29755,29756,29757,29761,29763,29764,29765,29768,29769,29771,
29772,29773,29784,29785,29791,29792,29793,29797,29799,29800,29801,29804,29816,29817,29828,29829,29832,29833,29836,29841,
29844,29848,29849,29855,29856,29868,29873,29876,29880,29881,29889,29892,29897,29916,29919,29920,29925,29929,29941,29944,
29945,29949,29952,29953,29961,29967,29971,29973,29975,29976,29977,29980,29983,29985,29987,29989,29992,29993,29996,29999,

This shows how well the holes are filled up,
but of course it's not a proof that it is indeed a permutation.

Maximilian

PS: if s.o. is interested in:
{t=0;last=1;for(n=1,49999,t+=1<<last/*;print1(last",")*/;
for(i=last\3+1,last\2,bittest(t,i)&next;last=i;next(2));
for(i=last*2,last*3-1,bittest(t,i)&next;last=i;next(2)); error("THE
END: n=",n))}
I checked it works up to 50 000.
The largest value used at a given iteration seems to be ~ 2n:
iter.(n)  [log2(t)]
10000    23860
20000    46492
40000    89660
50000   117948