OEIS/Collatz Story

Introduction

Collatz sequences (also called trajectories) are sequences of integer numbers > 0. For any start value > 0 the elements of the sequence are constructed with two simple rules:

1. Even numbers are halved.
2. Odd numbers are multiplied by 3 and then incremented by 1.

Since decades it is unknown whether the final cyle 4 - 2 - 1 is always reached for all start values. This problem is the Collatz conjecture, for which the english Wikipedia states:

It is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.

Straightforward visualizations of Collatz sequences show no obvious structure. The sequences for the first dozen of start values are rather short, but the sequence for 27 suddenly has 112 elements.

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
• Manfred Trümper: The Collatz Problem in the Light of an Infinite Free Semigroup. Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 p.

Collatz Graph

When all Collatz sequences are read backwards, they form the Collatz graph starting with 1, 2, 4, 8 ... . At each node m > 4 in the graph, the path from the root (4) can be continued

• always to m * 2, and
• to (m - 1) / 3 if m ≡ 1 mod 3.

The Collatz conjecture claims that the graphs contains all numbers, and that - except for the trivial, leading cycle 1 - 2 - 4 - 1 - 2 - 4 ... - it has the form of a tree (without cycles). We will not consider the trivial cycle, and we start the graph with node 4, the root. Moreover, another trivial type of path starts when m ≡ 0 mod 3. We call such a path a sprout, and it contains duplications only. A sprout must be added to the graph for any node divisible by 3, therefore we will not consider them for the moment.

Graph Operations

Following Trümper, we use abbreviations for the elementary operations which transform a node (element, number) in the Collatz graph according to the following table (T1):

Name	Mnemonic	Distance to root	Mapping	Condition
d	down	-1	m ↦ m / 2	m ≡ 0 mod 2
u	up	-1	m ↦ 3 * m + 1	(m ≡ 1 mod 2)
s := ud	spike	-2	m ↦ (3 * m + 1) / 2)	m ≡ 1 mod 2
δ	divide	+1	m ↦ (m - 1) / 3	m ≡ 1 mod 3
µ	multiply	+1	m ↦ m * 2	(none)
σ := δµ	squeeze	+2	m ↦ ((m - 1) / 3) * 2	m ≡ 1 mod 3

We will mainly be interested in the reverse mappings (denoted with greek letters) which move away from the root of the graph.

3-by-2 Replacement

The σ operation, applied to numbers of the form 6 * m - 2, has an interesting property:

(6 * (3 * n) - 2) σ = 4 * 3 * n - 2 =  6 * (2 * n) - 2

In other words, as long as m contains a factor 3, the σ operation maintains the form 6 * x - 2, and it replaces the factor 3 by 2 (it "squeezes" a 3 into a 2). In the opposite direction, the s operation replaces a factor 2 in m by 3.

Motivation: Patterns in sequences with the same length

A closer look at the Collatz sequences shows a lot of pairs of adjacent start values which have the same sequence length, for example (from OEIS A070165):

142/104: 142 d  71 u 214 d 107 u 322 d 161 u 484 d  242 d 121 u 364 ] 182, 91, ... 4, 2, 1
143/104: 143 u 430 d 215 u 646 d 323 u 970 d 485 u 1456 d 728 d 364 ] 182, 91, ... 4, 2, 1
           +1  *6+4    +1  *6+4    +1  *6+4    +1   *6+4  *6+2    +0    +0 ...

The third line tells how the second line could be computed from the first. Walking from right to left, the step pattern is:

δ µ µ δ µ δ µ δ µ
µ µ δ µ δ µ δ µ δ

The alternating pattern of operations can be continued to the left with 4 additional pairs of steps:

 q? u [ 62 d  31 u  94 d  47 u 142 d ...
126 d [ 63 u 190 d  95 u 286 d 143 u ...
        +1  *6+4    +1  *6+4    +1

The pattern stops here since there is no number q such that q * 3 + 1 = 62.

Segment Construction

These patterns lead us to the construction of special subsets of paths in the Collatz graph which we call segments. They lead away from the root, and they always start with a node m ≡ -2 mod 6. Then they split and follow two subpaths in a prescribed sequence of operations. The segment construction process is stopped when the next node in one of the two subpaths becomes divisible by 3, resp. when a δ operation is no more possible. We assemble the segments as rows of an infinite array C[i,j], the so-called segment directory.

Informally, and in the two examples above, we consider the terms betweeen the square brackets. For the moment, we only take those which are which are ≡ 4 mod 6 (for "compressed" segments, below there are also "detailed" segments where we take all). We start at the right and with the lower line, and we interleave the terms ≡ 4 mod 6 of the two lines to get a segment.

The columns in one row i of the array C are constructed as described in the following table (T2):

Column j	Operation	Formula	Condition	Sequence
1		6 * i - 2		4, 10, 16, 22, 28, ...
2	C[i,1] µµ	24 * (i - 1) + 16		16, 40, 64, 88, 112, ...
3	C[i,1] δµµ	24 * (i - 1) / 3 + 4	i ≡ 1 mod 3	4, 28, 52, 76, 100, ...
4	C[i,2] σ	48 * (i - 1) / 3 + 10	i ≡ 1 mod 3	10, 58, 106, 134, ...
5	C[i,3] σ	48 * (i - 7) / 9 + 34	i ≡ 7 mod 9	34, 82, 130, 178, ...
6	C[i,2] σσ	96 * (i - 7) / 9 + 70	i ≡ 7 mod 9	70, 166, 262, 358, ...
7	C[i,3] σσ	96 * (i - 7) / 27 + 22	i ≡ 7 mod 27	22, 118, 214, 310, ...
8	C[i,2] σσσ	192 * (i - 7) / 27 + 46	i ≡ 7 mod 27	46, 238, 430, 622, ...
9	C[i,3] σσσ	192 * (i - 61) / 81 + 142	i ≡ 61 mod 81	142, 334, ...
...	...	...	...	...

The first column(s) C[i,1] will be denoted as the left side of the segment (or of the whole directory), while the columns C[i,j], j > 1 will be the right part. The first few lines of the segment directory are the following:

	1	2	3	4	5	6	7	8	9	10	11	...	2*j	2*j+1
i	6*i‑2	µµ	δµµ	µµσ	δµµσ	µµσσ	δµµσσ	µµσ3	δµµσ3	µµσ4	δµµσ4	...	µµσj-1	δµµσj-1
1	4	16	4	10
2	10	40
3	16	64
4	22	88	28	58
5	28	112
6	34	136
7	40	160	52	106	34	70	22	46

There is a more elaborated segment directory with 5000 rows.

Properties of the segment directory

We make a number of claims for segments:

• (C1) All nodes in the segment directory are of the form 6 * n - 2.

This follows from the formula for columns C[i,1..3], and for any higher column numbers from the 3-by-2 replacement property of the σ operation.

• (C2) All segments have a finite length.

At some point the σ operations will have replaced all factors 3 by 2.

• (C3) All nodes in the right part of a segment have the form 6 * (3ⁿ * 2^m * f) - 2 with the same "3-2-free" factor f.

This follows from the operations for columns C[i,1..3], and from the fact that the σ operation maintains this property.

• (C4) All nodes in the right part of a particular segment are
  • different among themselves, and
  • different from the left side of that segment (except for the first segment for the root 4).

For C[i,1..2] we see that the values modulo 24 are different. For the remaining columns, we see that the exponents of the factors 2 and 3 are different. They are shifted by the σ operations, but they alternate, for example (in the segment with left part 40):

160 = 6 * (33 * 20 * 1) - 2
 52 = 6 * (32 * 20 * 1) - 2
106 = 6 * (32 * 21 * 1) - 2
 34 = 6 * (31 * 21 * 1) - 2
 70 = 6 * (31 * 22 * 1) - 2
 22 = 6 * (30 * 22 * 1) - 2
 46 = 6 * (30 * 23 * 1) - 2

• (C5) There is no cycle in a segment (except for the first segment for the root 4).

Segment Lengths

The segment directory is obviously very structured. The lengths of the compressed segments follow the pattern

4 2 2 4 2 2 L1 2 2 4 2 2 4 2 2 L2 2 2 4 2 2 ...

with two fixed lengths 2 and 4 and some variable lengths L₁, L₂ ... > 4. For the left parts 4, 40, 364, 3280, 29524 (OEIS A191681), the segment lengths have high values 4, 8, 12, 16, 20 which did not occur before. Those left parts are (9ⁿ⁺¹ - 1) / 2, or 4 * Sum(9ⁱ, i = 0..n).

Coverage of the Right Part

We now examine the modular conditions which result from the segment construction table in order to find out how the numbers of the form 6 * n - 2 are covered by the right part of the segment directory, as shown in the following table (T3):

Columns j	Covered	Remaining
2-3	4, 16 mod 24	10, 22, 34, 46 mod 48
3-4	10, 34 mod 48	22, 46, 70, 94 mod 96
5-6	70, 22 mod 96	46, 94, 142, 190 mod 192
7-8	46, 142 mod 192	94, 190, 286, 382 mod 384
...	...	...

We can always exclude the first and the third element remaining so far by looking in the next two columns of segments with sufficient length.

• (C6) There is no limit on the length of a segment.

We only need to take a segment which, in its right part, has a factor of 3 with a sufficiently high power, and the σ operations will stretch out the segment accordingly.

Therefore we can continue the modulus table above indefinitely, which leads us to the claim:

• (C7) All numbers of the form 6 * n - 2 occur exactly once in the right part of the segment directory, and once as a left side. There is a bijective mapping between the left sides and the elements of the right parts.

The sequences defined by the columns in the right part all have different modulus conditions. Therefore they are all disjoint. The left sides are disjoint by construction.

Attachment Directory

So far - by the segments - we possess an infinite set of small - cyclefree - trees with disjoint nodes and two branches. We now want to connect them to a single big tree which will later become the backbone of the Collatz tree.

Parallel to the segment directory we maintain a so-called attachment directory A which only contains a flag telling

• whether the tree corresponding to segment was already attached to the tree of some other segment, or
• whether it is still unattached.

Initially all trees are unattached.

Tree attachment rules

We attach a set of source rows (segments, trees) in C to appropriate target rows and columns by starting a gedankenexperiment analogous to Hilbert's hotel.

We operate on A as follows. Considering simultaneously all source rows i > 1 - omitting the root segment - which are so far unattached, and which fulfill some modularity condition (the source row set), we attach (or identify, connect) their trees in parallel to the unique occurrences of their left side in the right part of C (target rows and target column). Afterwards, we set the flag of those source rows/segments in A to attached, and we do not try to attach those segments anymore.

The following table (T4) shows the unique position where source rows must be attached to target rows and columns:

Source rows i	First source rows	Target rows	Target column	First target rows	New pos.	Remaining rows	Remaining Fraction
i ≡ 3 mod 4	3, 7, 11, 15 ...	30 * (i - 3) / 4 + 1	2	1, 2, 3, 4 ...	<	i ≡ 0, 1, 2 mod 4	3/4
i ≡ 1 mod 4	(1), 5, 9, 13 ...	31 * (i - 1) / 4 + 1	3	(1), 4, 7, 10 ...	<(=)	i ≡ 0, 2, 4, 6 mod 8	1/2
i ≡ 2 mod 8	2, 10, 18, 26 ...	31 * (i - 2) / 8 + 1	4	1, 4, 7, 10 ...	<	i ≡ 0, 4, 6 mod 8	3/8
i ≡ 6 mod 8	6, 14, 22, 30 ...	32 * (i - 6) / 8 + 7	5	7, 16, 25, 34 ...	>	i ≡ 0, 4, 8, 12 mod 16	1/4
i ≡ 12 mod 16	12, 28, 44, 60 ...	32 * (i - 12) / 16 + 7	6	7, 16, 25, 34 ...	<	i ≡ 0, 4, 8 mod 16	3/16
i ≡ 4 mod 16	4, 20, 36, 52 ...	33 * (i - 4) / 16 + 7	7	7, 34, 61, 88 ...	>	i ≡ 0, 8, 16, 24 mod 32	1/8
i ≡ 8 mod 32	8, 40, 72, 104 ...	33 * (i - 8) / 32 + 7	8	7, 34, 61, 88 ...	<	i ≡ 0, 16, 24 mod 32	3/32
i ≡ 24 mod 32	24, 56, 88, 120 ...	34 * (i - 24) / 32 + 61	9	61, 142, 223, 304 ...	>	i ≡ 0, 16, 32, 48 mod 64	1/16
i ≡ 48 mod 64	48, 112, 176, 240 ...	34 * (i - 48) / 64 + 61	10	61, 142, 223, 304 ...	>	i ≡ 0, 16, 32 mod 64	3/64
i ≡ 16 mod 64	16, 80, 144, 208 ...	35 * (i - 16) / 64 + 61	11	61, 304, 547, 790 ...	>	i ≡ 0, 32, 64, 96 mod 128	1/32
...	...	...		...	>	...	...

It should be obvious how the next rows of this infinite table must be filled:

• The residues of 2^k in the first column are 3 * 2^k-2, 1 * 2^k-2 in an alternating sequence.
• The additive constants in the second column are the indexes of the variable length segments with left parts (4), 40, 364, 3280, 29524 (OEIS A191681) mentioned above. They are repeated 4 times since the corresponding lengths "jump" by 4.
• (C8) The odd source rows are covered by rules 2 and 3. All even source rows are covered by rules 4 and following.

We identify and denote the attachment rules with the - bold - target column number. The construction is such that the target column always exists.

• (C9) It does not matter in which order the single attachment steps are performed.

Even if, during the application of one rule, a source row becomes a target row or vice versa, the target row and column is still found, and the attachment of the segment/tree can be performed.

• (C10) The target column depends on the modularity condition for i alone, but not on i.

This is provable from the graph operations (&delta, µ, σ) which are tied to the columns.

No Cycles

• (C11) The attachment process does not create any new cycle (in addition to the one in the root segment).

Let a segment/tree t₁ with left side n₁ and right part R₁ be attached to node n₁ in the right part R₂ of the unique segment/tree t₂ which has the left side by n₂. t₁ and t₂ are disjoint trees by (C4), therefore the result of such a single attachment step is a tree again (u₂, still with left side n₂).

Reduction of T4

We will now use the rules of T4 to reduce the set of unattached segments in A.

Contraction of pairs

The target rows for the rules 5 and 6 are the same (and also for 7 and 8, 9 and 10, etc.) We may safely apply rules 5, 7, 9 and all following odd rules. We are left with the source rows for which the rules 2, 3, 4 and all following even numbers are applicable.

Rules > 3 attach to rows ≡ 1 mod 3

We see that the target rows for the rules with numbers > 3 are all ≡ 1 mod 3. We may safely apply all these rules (with rule 4 we do not attach source row 1), and thereafter we are left with the odd source rows ≡ 1, 3 mod 4 only, for which rules 2, 3 are applicable. Both rules move down.

The Collatz Tree

• (C11) The remaining single tree is a subgraph of the Collatz graph.

The edges of the compressed tree carry combined operations µµ, δµµ and σ = δµ.

So far, numbers of the form x ≡ 0, 1, 2, 3, 5 mod 6 are missing from the compressed tree.

We insert intermediate nodes into the compressed tree by applying operations on the left parts of the segments as shown in the following table (T5):

Operation	Condition	Resulting Nodes	Remaining Nodes
δ		2 * i - 1	i ≡ 0, 2, 6, 8 mod 12
µ		12 * i - 4	i ≡ 0, 2, 6 mod 12
δµ	i ≡ 1, 2 mod 3	4 * i - 2	i ≡ 0, 12 mod 24
δµµ	i ≡ 2 mod 3	8 * i - 4	i ≡ 0 mod 24
δµµµ	i ≡ 2 mod 3	16 * i - 8	(none)

The first three rows in T5 care for the intermediate nodes at the beginning of the segment construction with columns 1, 2, 3. Rows 4 and 5 generate the sprouts (starting at multiples of 3) which are not contained in the segment directory.

We call such a construction a detailed segment (in contrast to the compressed segments described above).

A detailed segment directory can be created by the same Perl program. In that directory, the two subpaths of a segment are shown in two lines. Only the highlighted nodes are unique.

• (C11) The connectivity of the compressed tree remains unaffected by the insertions.
• (C12) With the insertions of T5, the compressed tree covers the whole Collatz graph.
• (C13) The Collatz graph is a tree (except for the trivial cycle 4-2-1).