I noticed something about the Collatz Conjecture, (I was literally obsessed with trying to prove it). I of course have NO intention of trying to prove it, clearly it is beyond my reach and I hope not to offend anyone by what may be a nonsensical observation, but I was a bit curious.
This is what I've noticed or "my conjecture". Pick any $\textbf{odd number}$ n. Then $4n+1$ takes exactly $+2$ more steps than $n$.
I also came across this $4n+1$ rule a lot of times in other observations, I'm not sure if this is any important or completely nonsensical. If you expand the set out you get:
$25, 101, 405, 1612$
$23, 93, 373, 1413$
$17, 69, 277, 1109$
$11, 45, 181, 725$
$9, 29, 117, 469, 1877$
$7, 29, 117, 469, 1877$
$3, 13, 53, 213$
$1, 5, 21, 85, 341$
I checked this for all values up to 2000001, so nothing concrete at all, but just a bit curious at any explanations for this pattern and (potentially) an explanation at a function to find the number of steps it takes for a odd number to reach 1?
The problem is that the number of steps it takes for the left-most number seems random...., thus making it impossible to determine the other number's behaviour without knowing the "basis".
If $n$ is odd, we have $4n+1 \to 12n+4 \to 6n+2 \to 3n+1$, while $n \to 3n+1$ so the streams join with $4n+1$ taking two extra steps.