The 3𝑥+1 Problem as a String Rewriting System

Table 1

3𝑥+1 path (odd integers) for 𝑛=27 expressed in integer and run notations.

No.	Integer	Run notation	No.	integer	Run notation
0	27	2 1 2	21	283	2 1 2 3 1
1	41	1 2 1 1 1	22	425	1 2 1 1 1 1 2
2	31	5	23	319	6 2 1
3	47	4 1 1	24	479	5 1 3
4	71	3 3 1	25	719	4 2 2 1 1
5	107	2 1 1 1 2	26	1079	3 1 2 4 1
6	161	1 4 1 1 1	27	1619	2 2 1 1 1 2 2
7	121	1 2 4	28	2429	1 1 5 1 1 2 1
8	91	2 1 2 1 1	29	911	4 3 3
9	137	1 2 1 3 1	30	1367	3 1 1 1 1 1 1 1 1
10	103	3 2 2	31	2051	2 9 1
11	155	2 1 2 2 1	32	3077	1 1 1 7 2
12	233	1 2 1 1 3	33	577	1 5 1 2 1
13	175	4 1 1 1 1	34	433	1 3 2 1 2
14	263	3 5 1	35	325	1 1 1 3 1 1 1
15	395	2 1 1 3 2	36	61	1 1 4
16	593	1 3 1 1 1 2 1	37	23	3 1 1
17	445	1 1 4 1 2	38	19	2 3 1
18	169	3 2 1 1 1	39	53	1 1 1 1 2
19	251	2 1 5	40	5	1 1 1
20	377	1 2 4 1 1	41	1	1