List of properties of the Syracuse sequence

Question

Where can I find an as exhaustive as possible list of all the properties (empirical or proven) related to the Collatz conjecture ?

For example I noticed that starting from $2^{n}-1$ the sequence always reach $3^{n}-1$ at some point : $$\forall n \in \mathbb{N}^{*} \quad \exists m \in \mathbb{N} \; \mid \; 2^{n}-1 \underset{m \; iterations}{\longrightarrow} 3^{n}-1$$ which is in fact trivial using base 2 (and also $m = 2n$), and wanted to see what are the other known properties, trivial or not, conjectured or proven.

Is there a place where such a list is kept, maybe even a regularly updated one and on the internet ?

Answer 1

The authoritative reference on the Collatz conjecture is the book The Ultimate Challenge: the 3x+1 problem, edited by Jeffrey C. Lagarias, the leading expert in the subject.

There is also the web page On The 3x + 1 Problem maintained by Eric Roosendaal.

Answer 2

In addition to the lhf's answer, the website of Jeffrey C. Lagarias and its An Annotated Bibliography in preprints section. Also its older article: The 3x+1 problem and its generalizations.

Answer 3

For HOTPO (Half OR Triple Plus One), let’s denote the number of half operations as $H$ and number of triple as $T$. Now suppose that for $2^{n-1}$, $H=T=n$ and $n$ interactions. If given a number like $A$, then we may write:

$$\frac{(2^n -1)3^n + A}{2^n}=(3^n-1).$$

So, $A =3^n-2^n$. Or $(2^n -1)3^n + 3^n-2^n =(3^n-1)2^n$.

This is an over simplified form of a formula that can be found by HOTPO on a number like $x$ as follows:

$$x_i =\frac{3^nx+ 3^n +\sum 3^{n-k_i} 2^{k_i} + \dots)+ 2^{\sum k_i}}{2^ {\sum k_i}}.$$ In our case $x = 2^n-1$ and $x_i=3^n-1$, where $k_i$ represents $H$ (half operation) in interactions. Therefore there exists at least one case in which $H=T=n$ and $x = 2^n-1$ that would give a number like $3^n -1$.

I checked this to be true for when $n$ is odd; ($n =2k+1$). In this case $m=n$, where $m$ is the number of interactions. You can check it for $n = 3, 5, 7, \dots$.

For example for $n=3$ we have:

$x=2^3-1 =7$; $3 \times 7+1=22$; $\frac{22}{2}=11$; $11 \times 3+1=34$; $\frac{34}{2}=17$; $17\times 3+1=52$; $\frac{52}{2}=26$; and $26=3^3-1$.

As can be seen, the number of triple operation is $3$($=n$) which is equal to number of half operations.

Answer 4

I can tell you there are only three functional properties for the odd numbers of a sequence:

$f(n) = \begin{cases} n+ \frac {n+x}{2} & \mbox{if } n+x \equiv 0 \mbox{ (mod } 4) \\ n-\frac{n-x}{4} & \mbox{if } n-x \equiv 0 \mbox{ (mod } 8) \\ \frac {n-\frac{n+x}{2}}{2} & \mbox{otherwise} \end{cases}$

where $x$ is $1$ (or some other value).

Algorithmically,

If $n+x$ is divisible by $4$, multiply by $1.5$, subtract $x$
If $n-x$ is divisible by $8$, multiply by $0.75$, add $x$
Else multiply $n-x$ by $0.25$