Take any number...

Take any number
if it is even, halve it;
if it is odd, multiply it by 3 and add 1 and then halve it.
repeat these operations until you reach the number 1.

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(2ⁿ) = n
e.g. 16 = 2⁴, so S(16)=4.
We can ignore the even numbers, by the way, as s(k x 2ⁿ) = s(k) + n

Table 1: Some results k<100)		Table 2: Numbers matched by number of links

n	S(n)	n	S(n)	n	S(n)
3	5	35	10	67	19
5	4	37	15	69	11
7	11	39	23	71	65
9	13	41	69	73	73
11	10	43	20	75	11
13	7	45	12	77	16
15	12	47	66	79	24
17	9	49	17	81	16
19	14	51	17	83	70
21	6	53	9	85	8
23	11	55	71	87	21
25	16	57	22	89	21
27	70	59	22	91	59
29	13	61	14	93	13
31	67	63	68	95	61
33	19	65	19	97	75

s(k)	k	s(k)	k
3	-	14	19, 61
4	5	15	37
5	3	16	25, 77, 81
6	21	17	49, 51
7	13	18	99
8	85	19	33, 65, 67
9	17, 53	20	43
10	11, 35	21	87, 89
11	7, 23, 69, 75	22	57, 59
12	15, 45	23	39
13	9, 29, 93	24	79

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