I'm new to all of this and I don't know if this is the right topic to open this discussion but something is bothering me and I need help:
 First of all why does replacing (3n+1) in the conjecture for (2n+2) seems to bring the same outcome (1); is it possible to prove this?
 Second of all when I look for specific things in the conjecture using python the results seem to always form patterns, for example, when searching for all the odd natural numbers that when multiplied by three and added one are equal to the power of two this function appeared:
(2^(2x+2)1)/3, x>0
so for x=1, the result is 5; (5*3)+1=16=2**4
for x=2, the result is 21; (21*39)+1=64=2**6
...and so on...
 And finally, recently it came to me the idea to use python to search for a ratio between the number of (n/2) steps used and (3n+1) steps. And then I had the idea to search for the lowest ratio! In other words I wanted to find all the odd numbers that had more 3n+1 steps and less 2/n steps. And it seems that this ratio converges to some number. Can anyone help me find which?
Just a quick example:
ratio=2.5; number being used=3; 5 (n/2 steps) / 2 (3n+1) steps = 2.5
2.2 7  11 / 5
2.16666666667 9  13 / 6
1.70731707317 27  70 / 41
1.69512195122 230631  278 / 164
1.68783068783 626331  319 / 189
1.68717948718 837799  329 / 195
1.68599033816 1723519  349 / 207
1.68468468468 3732423  374 / 222
1.68421052632 5649499  384 / 228
1.67741935484 6649279  416 / 248
1.67578125 8400511  429 / 256
1.65826330532 63728127  592 / 357
(I’m disregarding even numbers because they are just odd numbers multiplied by some term of the power of two)
Some help would be much appreciated. I'm just curious and trying to understand a bit more.
Thank you so much!
