I may be using the wrong terminology here, but please bear with me, I'm not a mathematician, just a hobbyist... ;-)
I noticed that for some pairs of numbers n and n+1, when you put them through the Collatz rule, sometimes they fall into step with each other. The rule - as I assume will be known - is: when n is even, divide by two, when n is odd, multiply by three and add 1. Repeat this until you get 1 as a result.
As an example: when you put the numbers 350 and 351 through this algorithm, they will have the same sequence after step 12. Like so:
350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, etc
351, 1054, 527, 1582, 791, 2374, 1187, 3562, 1781, 5344, 2672, 1336, 668, 334, etc
Other examples are: 242 and 243 (after only 5 steps), 1346 and 1347 (same), 237 and 238 (8 steps)
If the sequences converge it seems to happen after only a small number of steps.
Has any study been done on this phenomenom? Does anyone have a hint on where to start looking?