Permutation? Floor Of Ratio Of Adjacent Terms = 2

[Robert Israel]

I used the following Maple code:

a[1]:= 1; Taken:= {1};
for i from 2 to 5000 do
    q1:= floor(a[i-1]/3+1);
    q2:= floor(a[i-1]/2);
    B:= {$q1 .. q2} minus Taken;
    if B <> {} then a[i]:= B[1]; Taken:= Taken union {B[1]}; next end if;
    q1:= 2*a[i-1];
    q2:= 3*a[i-1];
    B:= {$q1 .. q2} minus Taken;
    if B <> {} then a[i]:= B[1]; Taken:= Taken union {B[1]}; next end if;
    print("Stopped!"); break
  end do:

The first 200 members of the sequence are

1, 2, 4, 8, 3, 6, 12, 5, 10, 20, 7, 14, 28, 11, 22, 9, 18, 36, 13, 26, 52, 
19, 38, 15, 30, 60, 21, 42, 16, 32, 64, 23, 46, 17, 34, 68, 24, 48, 96, 
33, 66, 25, 50, 100, 35, 70, 27, 54, 108, 37, 74, 29, 58, 116, 39, 78, 31, 
62, 124, 43, 86, 40, 80, 160, 55, 110, 41, 82, 164, 56, 112, 44, 88, 176, 
59, 118, 45, 90, 180, 61, 122, 47, 94, 188, 63, 126, 49, 98, 196, 67, 134, 
51, 102, 204, 69, 138, 53, 106, 212, 71, 142, 57, 114, 228, 77, 154, 65, 
130, 260, 87, 174, 72, 144, 288, 97, 194, 73, 146, 292, 99, 198, 75, 150, 
300, 101, 202, 76, 152, 304, 103, 206, 79, 158, 316, 107, 214, 81, 162, 
324, 109, 218, 83, 166, 332, 111, 222, 84, 168, 336, 113, 226, 85, 170, 
340, 115, 230, 89, 178, 356, 119, 238, 91, 182, 364, 123, 246, 92, 184, 
368, 125, 250, 93, 186, 372, 127, 254, 95, 190, 380, 128, 256, 104, 208, 
416, 139, 278, 105, 210, 420, 141, 282, 117, 234, 468, 157, 314, 120, 240, 
480, 161

I can tell you that the a[n] exist at least up to n=5000. 
The least integer that hadn't yet been in the sequence to that point is
2805.  It looks to me like the sequence will continue forever and that
it will be a permutation of the positive integers.

Cheers,
Robert Israel

On Mon, 7 Apr 2008, Leroy Quet wrote:

> First, thanks to everyone who contributed to the
> last discussion I started.
>
> Now a new topic.
>
> I just submitted this:
>
> %S A139080 1,2,4,8,3,6,12,5,10,20,7,14,28,11,22,9
> %N A139080 a(1)=1. a(n) = the smallest positive
> integer not occurring earlier in the sequence
> such that
> floor(max(a(n),a(n-1))/min(a(n),a(n-1))) = 2.
> %C A139080 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?
> %O A139080 1
> %K A139080 ,more,nonn,
>
> (I find the sequence to be more beautiful than is
> suggested at first glance by its definition.)
>
> The question I have is, as seen in the comment
> line:
>
> 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?
>
> And of course I must ask, did I make a mistake
> with the terms I give?
>
> Thanks,
> Leroy Quet
>
>
>
>
>      ____________________________________________________________________________________
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>