Loops in other systems

Introduction

On other pages we've seen how the 's' values, the potential numerators found while searching for loops in 3x+1, are actually loop seeds in other systems (like 3x+5). Mostly though, these 's' values are seeds in systems with very large 'k' values. Loops in systems with small 'k' values usually derive from systems where 's' and 'K' have a common factor (f), so k = K∕f and the seed is s∕f.

I've ignored systems where k is a multiple of 3 as they aren't covered by the equations considered on this site (as 2^m−3ⁿ can never be a multiple of 3).

I ran a search for cycles in systems with k values up to 200 (ignoring multiples of 2 and 3) and seed values up to 100,000,000. The largest seed I actually found was 35,275 in 3x+145 and the largest cycle had 80 members, in 3x+143.

It looks like most if not all cycles are in the domain where s/k is relatively small. 's' is the smallest member of the loop, other members can be relatively large.

Loops in small k values

This section shows some of the loops discovered in systems with k values up to 19. The largest loop is 18 in 3x+17, the largest seed is 347 in 3x+5, and the largest s∕k ratio is 69.4 in 3x+5.

Loops in 3x+1

Seed	Size	Seed/K	Loop
1	1	1.0	[1, 1]

Loops in 3x+5

Seed	Size	Seed/K	Loop
1	1	0.2	[1, 1]
5	1	1.0	[5, 5]
19	3	3.8	[19, 31, 49, 19]
23	3	4.6	[23, 37, 29, 23]
187	17	37.4	[187, 283, 427, 643, 967, 1453, 1091, 1639, 2461, 1847, 2773, 2081, 781, 587, 883, 1327, 1993, 187]
347	17	69.4	[347, 523, 787, 1183, 1777, 667, 1003, 1507, 2263, 3397, 2549, 1913, 359, 541, 407, 613, 461, 347]

Loops in 3x+7

Seed	Size	Seed/K	Loop
5	2	0.7	[5, 11, 5]
7	1	1.0	[7, 7]

Loops in 3x+11

Seed	Size	Seed/K	Loop
1	2	0.1	[1, 7, 1]
11	1	1.0	[11, 11]
13	8	1.2	[13, 25, 43, 35, 29, 49, 79, 31, 13]

Loops in 3x+13

Seed	Size	Seed/K	Loop
1	1	0.1	[1, 1]
13	1	1.0	[13, 13]
131	15	10.1	[131, 203, 311, 473, 179, 275, 419, 635, 959, 1445, 1087, 1637, 1231, 1853, 1393, 131]
211	5	16.2	[211, 323, 491, 743, 1121, 211]
227	5	17.5	[227, 347, 527, 797, 601, 227]
251	5	19.3	[251, 383, 581, 439, 665, 251]
259	5	19.9	[259, 395, 599, 905, 341, 259]
283	5	21.8	[283, 431, 653, 493, 373, 283]
287	5	22.1	[287, 437, 331, 503, 761, 287]
319	5	24.5	[319, 485, 367, 557, 421, 319]

Loops in 3x+17

Seed	Size	Seed/K	Loop
1	2	0.1	[1, 5, 1]
17	1	1.0	[17, 17]
23	18	1.4	[23, 43, 73, 59, 97, 77, 31, 55, 91, 145, 113, 89, 71, 115, 181, 35, 61, 25, 23]

Loops in 3x+19

Seed	Size	Seed/K	Loop
5	5	0.3	[5, 17, 35, 31, 7, 5]
19	1	1.0	[19, 19]

Larger 'k' values

This table summarises the results up to k=199.

K	Seed	Size	Seed/K
1	1	1	1.0
5	1	1	0.2
5	5	1	1.0
5	19	3	3.8
5	23	3	4.6
5	187	17	37.4
5	347	17	69.4
7	5	2	0.7
7	7	1	1.0
11	1	2	0.1
11	11	1	1.0
11	13	8	1.2
13	1	1	0.1
13	13	1	1.0
13	131	15	10.1
13	211	5	16.2
13	227	5	17.5
13	251	5	19.3
13	259	5	19.9
13	283	5	21.8
13	287	5	22.1
13	319	5	24.5
17	1	2	0.1
17	17	1	1.0
17	23	18	1.4
19	5	5	0.3
19	19	1	1.0
23	5	2	0.2
23	7	2	0.3
23	23	1	1.0
23	41	26	1.8
25	5	1	0.2
25	7	8	0.3
25	17	4	0.7
25	25	1	1.0
25	95	3	3.8
25	115	3	4.6
25	935	17	37.4
25	1735	17	69.4
29	1	1	0.0
29	11	9	0.4
29	29	1	1.0
29	3811	41	131.4
29	7055	41	243.3
31	13	12	0.4
31	31	1	1.0
35	7	1	0.2
35	13	4	0.4
35	17	4	0.5
35	25	2	0.7
35	35	1	1.0
35	133	3	3.8
35	161	3	4.6
35	1309	17	37.4
35	2429	17	69.4
37	19	3	0.5
37	23	3	0.6
37	29	3	0.8
37	37	1	1.0
41	1	8	0.0
41	41	1	1.0
43	1	3	0.0
43	43	1	1.0
47	5	7	0.1
47	25	16	0.5
47	47	1	1.0
47	65	4	1.4
47	73	4	1.6
47	85	4	1.8
47	89	4	1.9
47	101	4	2.1
49	25	22	0.5
49	35	2	0.7
49	49	1	1.0
53	53	1	1.0
53	103	17	1.9
55	1	4	0.0
55	5	2	0.1
55	7	2	0.1
55	11	1	0.2
55	41	16	0.7
55	55	1	1.0
55	65	8	1.2
55	209	3	3.8
55	253	3	4.6
55	2057	17	37.4
55	3817	17	69.4
59	1	11	0.0
59	59	1	1.0
59	133	6	2.3
59	149	6	2.5
59	181	6	3.1
59	185	6	3.1
59	217	6	3.7
59	221	6	3.7
61	1	1	0.0
61	61	1	1.0
61	235	41	3.9
65	5	1	0.1
65	13	1	0.2
65	19	12	0.3
65	65	1	1.0
65	247	3	3.8
65	299	3	4.6
65	655	15	10.1
65	1055	5	16.2
65	1135	5	17.5
65	1255	5	19.3
65	1295	5	19.9
65	1415	5	21.8
65	1435	5	22.1
65	1595	5	24.5
65	2431	17	37.4
65	4511	17	69.4
67	17	16	0.3
67	67	1	1.0
71	29	5	0.4
71	31	5	0.4
71	71	1	1.0
71	2585	17	36.4
71	2809	17	39.6
71	3985	17	56.1
71	4121	17	58.0
71	4409	17	62.1
73	5	4	0.1
73	19	32	0.3
73	47	8	0.6
73	73	1	1.0
77	1	16	0.0
77	7	2	0.1
77	55	2	0.7
77	77	1	1.0
77	91	8	1.2
79	1	20	0.0
79	7	20	0.1
79	79	1	1.0
79	233	14	2.9
79	265	14	3.4
83	65	14	0.8
83	83	1	1.0
83	109	7	1.3
83	157	7	1.9
85	5	2	0.1
85	7	56	0.1
85	17	1	0.2
85	85	1	1.0
85	115	18	1.4
85	323	3	3.8
85	391	3	4.6
85	3179	17	37.4
85	5899	17	69.4
89	17	8	0.2
89	89	1	1.0
91	1	24	0.0
91	7	1	0.1
91	25	6	0.3
91	59	6	0.6
91	65	2	0.7
91	91	1	1.0
91	917	15	10.1
91	1477	5	16.2
91	1589	5	17.5
91	1757	5	19.3
91	1813	5	19.9
91	1981	5	21.8
91	2009	5	22.1
91	2233	5	24.5
95	1	36	0.0
95	17	5	0.2
95	19	1	0.2
95	23	5	0.2
95	25	5	0.3
95	95	1	1.0
95	361	3	3.8
95	437	3	4.6
95	3553	17	37.4
95	6593	17	69.4
97	1	6	0.0
97	13	3	0.1
97	97	1	1.0
101	7	6	0.1
101	11	6	0.1
101	19	3	0.2
101	23	3	0.2
101	29	3	0.3
101	31	3	0.3
101	37	3	0.4
101	101	1	1.0
103	5	8	0.0
103	23	22	0.2
103	103	1	1.0

K	Seed	Size	Seed/K
107	1	53	0.0
107	107	1	1.0
109	19	30	0.2
109	109	1	1.0
113	1	12	0.0
113	113	1	1.0
115	13	13	0.1
115	17	18	0.1
115	23	1	0.2
115	25	2	0.2
115	35	2	0.3
115	115	1	1.0
115	205	26	1.8
115	437	3	3.8
115	529	3	4.6
115	4301	17	37.4
115	7981	17	69.4
119	1	42	0.0
119	5	2	0.0
119	7	2	0.1
119	11	2	0.1
119	23	18	0.2
119	85	2	0.7
119	119	1	1.0
119	125	14	1.1
119	161	18	1.4
121	5	18	0.0
121	11	2	0.1
121	19	46	0.2
121	121	1	1.0
121	143	8	1.2
125	1	1	0.0
125	25	1	0.2
125	35	8	0.3
125	47	19	0.4
125	85	4	0.7
125	125	1	1.0
125	143	19	1.1
125	475	3	3.8
125	575	3	4.6
125	899	74	7.2
125	4675	17	37.4
125	8675	17	69.4
127	1	18	0.0
127	41	18	0.3
127	127	1	1.0
131	13	13	0.1
131	17	13	0.1
131	23	26	0.2
131	131	1	1.0
133	11	18	0.1
133	35	5	0.3
133	59	24	0.4
133	95	2	0.7
133	133	1	1.0
137	1	4	0.0
137	41	40	0.3
137	137	1	1.0
137	503	18	3.7
137	743	18	5.4
137	967	18	7.1
139	11	74	0.1
139	139	1	1.0
143	7	80	0.0
143	11	1	0.1
143	13	2	0.1
143	17	7	0.1
143	29	7	0.2
143	143	1	1.0
143	169	8	1.2
143	1441	15	10.1
143	2321	5	16.2
143	2497	5	17.5
143	2761	5	19.3
143	2849	5	19.9
143	3113	5	21.8
143	3157	5	22.1
143	3509	5	24.5
145	1	2	0.0
145	5	1	0.0
145	23	10	0.2
145	29	1	0.2
145	47	18	0.3
145	55	9	0.4
145	145	1	1.0
145	551	3	3.8
145	667	3	4.6
145	5423	17	37.4
145	10063	17	69.4
145	19055	41	131.4
145	35275	41	243.3
149	19	33	0.1
149	149	1	1.0
149	667	26	4.5
151	41	40	0.3
151	113	10	0.7
151	151	1	1.0
155	1	18	0.0
155	31	1	0.2
155	65	12	0.4
155	155	1	1.0
155	589	3	3.8
155	713	3	4.6
155	5797	17	37.4
155	10757	17	69.4
157	13	12	0.1
157	157	1	1.0
161	11	6	0.1
161	19	6	0.1
161	25	22	0.2
161	35	2	0.2
161	49	2	0.3
161	79	16	0.5
161	115	2	0.7
161	161	1	1.0
161	287	26	1.8
163	11	49	0.1
163	17	36	0.1
163	163	1	1.0
167	13	20	0.1
167	95	9	0.6
167	127	9	0.8
167	167	1	1.0
169	11	9	0.1
169	13	1	0.1
169	17	23	0.1
169	169	1	1.0
169	1703	15	10.1
169	2743	5	16.2
169	2951	5	17.5
169	3263	5	19.3
169	3367	5	19.9
169	3679	5	21.8
169	3731	5	22.1
169	4147	5	24.5
173	7	14	0.0
173	37	7	0.2
173	173	1	1.0
175	29	12	0.2
175	35	1	0.2
175	49	8	0.3
175	53	8	0.3
175	65	4	0.4
175	73	4	0.4
175	85	4	0.5
175	89	4	0.5
175	101	4	0.6
175	103	4	0.6
175	119	4	0.7
175	125	2	0.7
175	143	4	0.8
175	175	1	1.0
175	665	3	3.8
175	805	3	4.6
175	6545	17	37.4
175	12145	17	69.4
179	5	62	0.0
179	179	1	1.0
181	11	40	0.1
181	23	10	0.1
181	55	10	0.3
181	181	1	1.0
185	1	36	0.0
185	37	1	0.2
185	95	3	0.5
185	115	3	0.6
185	145	3	0.8
185	185	1	1.0
185	703	3	3.8
185	851	3	4.6
185	6919	17	37.4
185	12839	17	69.4
187	5	12	0.0
187	11	2	0.1
187	17	2	0.1
187	43	12	0.2
187	187	1	1.0
187	221	8	1.2
187	253	18	1.4
191	47	24	0.2
191	191	1	1.0
191	961	25	5.0
191	2405	25	12.6
193	1	29	0.0
193	5	5	0.0
193	31	12	0.2
193	91	50	0.5
193	193	1	1.0
197	5	69	0.0
197	197	1	1.0
199	13	6	0.1
199	47	6	0.2
199	199	1	1.0

Largest Seed

Searching k values up to 199 found a 41 member loop in 3x+145. This loop had the largest seed of those discovered in this search.

35,275	52,985	39,775	59,735	89,675	134,585	100,975
151,535	227,375	341,135	511,775	767,735	1,151,675	1,727,585
1,295,725	485,915	728,945	546,745	410,095	615,215	922,895
1,384,415	2,076,695	3,115,115	4,672,745	3,504,595	5,256,965	492,845
184,835	277,325	104,015	156,095	234,215	351,395	527,165
197,705	148,315	222,545	166,945	125,245	46,985	35,275

145 and 35275 have a common factor (5), so there is also a 41 member loop in 3x+29 (145/5) which has the seed s based on 7055 (35275/5). These are essentially the same loop and have the same s/k value, see the next section.

Record Holders

In an earlier section we observed the value of s/k for systems up to 3x+199. We also noticed that the largest loop seed found was in the 3x+145 system. If we look at the more interesting s/k value (it gives the same value for the essentially identical loops found in 3x+29 and 3x+145) we find the largest value so far is in 3x+29.

System	Seed	Size	Seed/K
3x+1	1	1	1.0
3x+5	347	17	69.4
3x+29	7055	41	243.3

This particular record isn't an easy one to calculate. To produce the above values I had to run 100 million calculations for 67 different systems. Even then I don't know if I've missed any loops, there may be a ridiculously large loop buried somewhere in 3x+1, for instance. It does look though that any loops there are are either close to 'k' or very large. I've not found any intermediate sized loops.

One way of scanning large numbers of systems would be to just choose a few numbers and see where they end up. '1' might be a good starting point and we can scan a large number of systems quite quickly. The corresponding results are.

System	Seed	Size	Seed/K
3x+1	1	1	1.0
3x+23	41	26	1.8
3x+53	103	17	1.9
3x+191	961	25	5.0

This hasn't worked very well, possibly because a seed of 1 tends to find the smaller loops. How about trying a big number, say 1000001? This finds the 131 loop in 3x+13 giving a slightly higher record of 10.1, but still nowhere near the real record of 243.3.

Convergence distribution

There don't seem to be any shortcuts for finding the loop record holders. Maybe this is because the majority of seeds converge on one of the smaller loops. Lest test this hypothesis in the 3x+29 system which has 5 loops including the record holder based on 7055.

First let's try all the seeds less than 7055. The results are shown in the table on the left. The table on the right shows the same test using the first 10 million odd numbers. Only about 0.28% converge on 7055.

Loop	Frequency
1	291
11	3101
29	122
3811	10
7055	3

Loop	Frequency
1	794171
11	8793106
29	344828
3811	39628
7055	28267