Collatz Streetmap

Introduction

When all Collatz sequences are read backwards, they form a graph starting with 1, 2 ..., hopefully without cycles (except for 1,2,4,1,2,4 ...). At each node n in the graph, the path starting at the root (4) and with the last node n can in principle be continued to 2 new nodes by a

• "m"-step: n * 2 (which is always possible), or a
• "d"-step: (n - 1) / 3 (which is possible only if n ≡ 1 mod 3).

When n mod 3 = 0, the path will continue with m-steps only, since the duplication maintains the divisibility by 3.

References

• Jeffry C. Lagarias, Ed.: The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010, ISBN 978-8218-4940-8. MBK78
• OEIS A07165: File of first 10K Collatz sequences, ascending start values, with lengths
• Gottfried Helms: The Collatz-Problem. A view into some 3x+1-trees and a new fractal graphic representation. Univ. Kassel.
• Klaus Brennecke: Collatzfolgen und Schachbrett, on Wikibooks

Motivation: Patterns in sequences with same length

When Collatz sequences are investigated, there are a lot of pairs of adjacent start values with the same sequence length, and with a characteristical neighbourhood of every other value, for example (from OEIS A070165):

142/104: [142 m  71 d 214 m 107 d 322 m 161 d 484 m  242 m 121 d | 364, 182, 91, ... 10, 5, 16, 8, 4, 2, 1]
           +1  *6+4    +1  *6+4    +1  *6+4    +1   *6+4  *6+2      =    =  ...
143/104: [143 d 430 m 215 d 646 m 323 d 970 m 485 d 1456 m 728 m | 364, 182, 91, ... 10, 5, 16, 8, 4, 2, 1]

Collatz roads

We define a "road" (with 2 parallel "lanes") as a sequence of pairs of elements (in 2 Collatz sequences with adjacent start values, read from right to left). A road is constructed by taking some n (364 in the example, the last common element of the 2 sequences) with n ≡ 4 mod 6, and by applying the steps

d m m d m d m d ... 
m m d m d m d m ...

in alternating sequence, until one of the elements in the pairs becomes divisible by 3. The construction yields one road with an upper lane (left elements of the pairs) and a lower lane (right elements), for example:

364: (121,728), (242,1456), (484,485), (161,970), (322,323), (107,646), (214,215), (71,430), (142,143) ...

The elements which are ≡ 4 mod 6 are emphasized. The construction continues until 63 ≡ 0 mod 3:

... (47,286), (94,95), (31,190), (62,63).

For better handling (e.g. in Excel), the roads for all starting values 4, 10, 16, 22 ... 4+6*n are listed as rows in an array:

r0	r1	r2	r3	r4	r5	r6	r7	r8	r9	r10	r11	r12	r13	r14	r15	...
start	len	dr0	mr0	mr2	mr3	mr4	dr5	dr6	mr7	mr8	dr9	dr10	mr11	mr12	dr13	...
Δ6		Δ2	Δ12	Δ4	Δ24	Δ8	Δ8	3Δ8	3Δ48	3Δ16	3Δ16	9Δ16	9Δ96	9Δ32	9Δ32	...
4	5	1	8	2	16	4	5	1	10	2	3
10	3	3	20	6	40	12	13
16	3	5	32	10	64	20	21
22	4	7	44	14	88	28	29	9	58
28	3	9	56	18	112	36	37
34	3	11	68	22	136	44	45
40	9	13	80	26	160	52	53	17	106	34	35	11	70	22	23	7	46	14	15
46	3	15	92	30	184	60	61
52	3	17	104	34	208	68	69
58	5	19	116	38	232	76	77	25	154	50	51
64	3	21	128	42	256	84	85
70	3	23	140	46	280	92	93
76	4	25	152	50	304	100	101	33	202

Observations

In contrast to the usual "chaotic" appearance of Collatz graphs, there is quite an amount of obvious structure in this construction: