[seqfan] A Collatz conjecture related sequence

[Ali Sada]

Hi everyone,

I would like to add the sequence below to the OEIS (if the editors think it’s worth it) and as usual, I need some help with the definition and the correct terms. 

1, 5, 13, 17, 53, 85, 113, 181, 241, 277, 301, 341, 401, 469, 565, 597, 625, 725, 833, 853, 965, 1109, 1205, 1285, 1477, 1621
If we apply the Collatz algorithm on any of these numbers (>1), we will land on a smaller number in the sequence. (Although some multiples of 3 technically follow the description above, they were excluded because no number would land on any of them when we apply the Collatz algorithm) 

For example, the Colltaz iterations of 9 are: 

9, 7, 11, 17, 13, 5, 1
What we want is to replace these numbers with their 4m+1 iterations, when necessary, to make them in descending order.

We don’t have problems with 1, 5, 13, or 17. 

So, we start by replacing 11 with a number larger than 17.
11*4+1 = 45. But 45 is a multiple of 3, so, we repeat
45*4+1 = 181
181 will replace 11 permanently.   

Now, we want to find an iteration of 7 that is larger than 181
7*4+1 = 29
29*4+1 = 117 
117*4+1 = 469  

Finally, we want to replace 9 with a number larger than 469
4*9+1 = 37
37*4+1 = 149
149*4+1= 597 (a multiple of 3, so we have to continue)
597*4+1 = 2389
Now, the Collatz numbers of 9 are in a fully descending order: 
2389, 469, 181, 17, 13, 5, 1

If we find a formula for the sequence above, and we prove (somehow) that all odd numbers have a “representation” in it, would that prove the Collatz conjecture? 
Best,

Ali