Collatz Conjecture

"Mathematics is not yet ready for such problems."
       — Paul Erdős

Consider positive integer sequences such that evens are followed by half the value, and, odds are followed by one more than triple the value. The Collatz conjecture states that all such sequences contain one:

2, 1, ...
3, 10, 5, 16, 8, 4, 2, 1, ...
4, 2, 1, ...
5, 16, 8, 4, 2, 1, ...
6, 3, 10, 5, 16, 8, 4, 2, 1, ...
7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, ...
8, 4, 2, 1, ...
9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, ...
10, 5, 16, 8, 4, 2, 1, ...
11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, ...
12, 6, 3, 10, 5, 16, 8, 4, 2, 1, ...
13, 40, 20, 10, 5, 16, 8, 4, 2, 1, ...
14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, ...
15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1, ...
16, 8, 4, 2, 1, ...
17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, ...
18, 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, ...
19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, ...
20, 10, 5, 16, 8, 4, 2, 1, ...
...etc.