Through messing around with some *Collatz values and trajectories, I came across values that seemed to rapidly increase and then 'rather quickly' fall back to one, like an explosion. (They all take less than 150 steps to reach one).
Here are the numbers I found interesting:
511
peak value in trajectory: 39364
takes 61 steps to reach 1
1023
peak value in trajectory: 118096
takes 62 steps to reach 1
32767
peak value in trajectory: 28697812
takes 129 steps to reach 1
65535
peak value in trajectory: 86093440
takes 130 steps to reach 1
Is this a well-known phenomenon? Why do the values seem to sharply fall back down to smaller values?
As a nice little bonus, the first and second pair's stopping times seem closely related for some reason...
*"Collatz" refers to the Collatz Conjecture. For a quick refresher:
If an integer $n$ is odd, then multiply by 3 and add 1. If $n$ is even, divide by 2. Repeat the process. The Collatz Conjecture states every positive starting number eventually reaches 1.
The property is long well known1). Per formula with some $A$ and odd $k$ $$a_1 = 2^A k -1 \to a_j = 3^A k -1$$ (with some $j$ depending on your definition of what is "one step")
The key is here that the value $-1$ is a fixpoint in the negative integers for the Collatz-transform.
If moreover $k=1$ then numbers of the form $3^A - 1$ have the primefactor $2$ to some easily to determine exponent: $$ \{3^A-1 ,2 \} = 1+ [A:2] + \{A,2\}$$ where the curly braces mean "valuation" (here the exponent of the primefactor $2$) and $[A:2]$ is a "Iverson-bracket" giving $1$ if $2 | A$ and $0$ else.
After that, $a_j = 3^{16} - 1 $ is followed by many $x/2$ steps (here $6$ steps): $$ \{3^{16}-1,2\} = 1 + [16:2] + \{16,2\} = 1+1+4 = 6 \\ a_j = 3^{16}-1 \to a_{j+6}={ 3^{16}-1 \over 2^6 }$$
You might be interested to look at the partial orbits of numbers of the forms $2^A k -5$,$2^A k -7$, $2^A k+1$ with some restrictions on the exponents $A$ ...
1) For instance I've seen it in an 1978-article by R. Crandall. But that's only the oldest original article I've found online - surely that property is so basic that it was known even earlier.