I looked at some of these questions for a college class. It was a class where we basically learned how to do math research by taking a really difficult math question, and basically running through the steps of research (using computer programs to get computational data, learning Latex to create a properly formatted final document with results, and then an hour long presentation of results). I can only get access to the pdf file of what we came up with on the university network for some reason, but I think I can remember some of the stuff we came up with. Our adviser is one of the leading people for the collatz conjecture (and keeps track of all the progress being made on all the variations of it), so I assume he would've mentioned something if we actually came up with something new/original/good.

One person in my group of 3 kind of focused on a different kind of generalized Collatz function. It was the real-valued extension where you use either cosines or sines to make a function that gives {3x+1 if odd, x/2 if even} in the most obvious fashion (since you can make a lot of functions that match the collatz function on integers).

My other friend did some stuff related to stopping time, like you were talking about. I think he was looking at some kind of ratio between x and the number of steps it took to get back to 1. We came up with some sort of computational number/estimate, and our adviser y had mentioned some guy had recently gotten a few more digits on what that maximum ratio of starting value/stopping time.

I worked on the generalization you're talking about, where you vary the parameters. I was limited on time (this was one of three projects for a 3 credit course out of a 17 credit schedule), so I didn't get as much done as I had wanted. I only came up with some data for a few of the smaller cases (after eliminating dumb cases, like 3x+2 where you get an increasing string of odd numbers), and I couldn't make any sense out of the data I got. Most of them had numbers reduce down to multiple different cycles (instead of all to 1,2,4 like in 3x+1), and there was seemingly no order to how many cycles or cycle length.

If you want come fun code to play with (assuming you don't come up with your own), the mathematica/maple/matlab files on the website for that course (

http://www.math.lsa.umich.edu/course...resources.html) are all about 3x+1, and if you have the software you could easily change some numbers to make it compute whatever generalization you want.