For example, starting with the number 5:
5 is odd; 5 x 3 = 15; add 1 : 16
16 ÷ 2 = 8 (step 1)
8 is even; halve it : 4 (step 2)
4 is even; halve it : 2 (step 3)
2 is even; halve it: 1, and stop. (step 4)
Taking it for granted that we will always reach the number 1 in the end, is there a connection between the number of steps required to reach the number 1 and the number we started with?
Here, if S(n) stands for the number of steps when we start with n, S(5) = 4.
So far all I've come up with is a pattern for the powers of two : S(2^{n}) = n
e.g. 16 = 2^{4}, so S(16)=4.
We can ignore the even numbers, by the way, as s(k x 2^{n}) = s(k) + n
Table 1: Some results k<100)  Table 2: Numbers matched by number of links 



That's as far as I've got to date.