All Questions
16 questions
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620
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Cocycles and the Collatz problem?
Let $T(n) = n+R(n)$, where $R(n) = -n/2 $ if $n\equiv 0 \mod 2$ else $R(n) = \frac{n+1}{2}$.
$R(n)$ is the Cantor ordering of the integers:
https://oeis.org/A001057
In the Collatz problem, one is ...
- 3,064
2
votes
1
answer
203
views
Finding a two point scrambled set for the function $g:[0,1] \rightarrow [0,1], x \mapsto \min_{n\in \mathbb{Z}} |3x-2n|$?
Let $I=[0,1]$ be the unit interval and $g$ as defined below.
Then $x \neq y$ with $x,y \in I$ are called "two point scrambled set"=$\{x,y\}$, if
$\lim\inf_{n \rightarrow \infty} | g^{(n)}(x)...
- 3,064
2
votes
2
answers
620
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A mutation of the Collatz disease
Given $k \in \mathbb N$, we define $f_k: \mathbb N \longrightarrow \mathbb N$ by
$$ f_k(x) = \begin{cases} \,\quad\dfrac{x}2 &\text{ if } x \text{ is even} \\\\ \dfrac{3x+3^k}{2} & \text{ if } ...
- 17.9k
4
votes
1
answer
2k
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Summary of “Almost All Orbits of the Collatz Map Attain Almost Bounded Values”
Terence Tao's 2019 paper ``Almost all Orbits of the Collatz map attain almost bounded values" is pretty famous. However, it's also long and complicated. I think there are useful techniques to ...
- 188
2
votes
0
answers
281
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Surreal numbers and the Collatz iteration as a game?
Let us define a game based on the Collatz function $C(n) = n/2$ if $n$ is even, otherwise $=3n+1$.
Each number $n$ represents a game played by left $L$ and right $R$:
$$n = \{L_n | R_n \}$$
The rules ...
- 3,064
11
votes
1
answer
1k
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Generating functions of Collatz iterates?
Let $C(n) = n/2$ if $n$ is even and $3n+1$ otherwise be the Collatz function.
We look at the generating function $f_n(x) = \sum_{k=0}^\infty C^{(k)}(n)x^k$ of the iterates of the Collatz function.
The ...
- 3,064
26
votes
3
answers
2k
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Unexpected behavior involving √2 and parity
This post makes a focus on a very specific part of that long post. Consider the following map:
$$f: n \mapsto \left\{
\begin{array}{ll}
\left \lfloor{n/\sqrt{2}} \right \rfloor & \...
11
votes
0
answers
809
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Borderline Collatz-like problems
The usual Collatz map is $C:n \mapsto n/2$ if $n$ even, $(3n+1)/2$ if $n$ odd. Let $f^{\circ (r+1)}:=f \circ f^{\circ r}$.
We suspect that for every fixed $n>0$, the sequence $C^{\circ r}(n)$ ...
6
votes
1
answer
562
views
How can I catalog these generalized Collatz problems?
The Collatz conjecture can be expressed in terms of a ruleset in the language $\{x,+,1,\rightarrow,;\}$:
$x + x + 1 \rightarrow x+x+x+1+1;$
$x + x \rightarrow x;$
Whenever a number matches the LHS ...
- 2,302
9
votes
1
answer
2k
views
A problem involving the inverse Collatz map
Let $C$ be the Collatz map on the natural numbers, defined by:
$$C(n) :=
\begin{cases}
n/2 & \text{if} \;n \;\text{even} \\
(3n+1)/2 & \text{if} \;n \;\text{odd}
\end{cases}$$
The inverse ...
1
vote
1
answer
502
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A Zsigmondy-theorem-analogy in the generalized Collatz-problem $3x+\rho$?
Remark : I've found a rather trivial answer for this question and so very likely the premise of paralleling it with the Zsigmondy-theorem is wrong, so this question might better be retracted. I'll ...
- 5,342
32
votes
2
answers
2k
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A Collatz-like problem on prime numbers
Consider the function $f$ on the prime numbers defined by $$ f(p):= \text{ the greatest prime factor of } 2p+1.$$ The iteration of $f$ from any prime $p<10^8$ converges to the cycle $$(3,7,5,11,23,...
6
votes
0
answers
448
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Are there always at least *five* divisions?
@JosephO'Rourke asked a question about a Collatz like function related to primes:
$f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is prime} \\
\lfloor n/2 \rfloor & \text{if} \;n \;\text{...
- 1,375
31
votes
4
answers
2k
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A Collatz-like function that bifurcates on primes
This is likely piling one mystery on another, but ...
I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows:
$$
f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is ...
- 151k
7
votes
1
answer
4k
views
Beyond Collatz: A $5n+1$ conjecture? [closed]
Let
$$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$
and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this ...
- 254
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1
answer
1k
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Implication for cycles (of some length $m$) in Collatz-type problems: typical ratio between largest and smallest element?
Background
Consider Collatz-type problems of the form $an + 1$, where $a > 2$ is a positive, odd integer (e.g., $3n + 1$, $5n +1 $, $7n + 1$, etc.). For convenience, automatically divide by two.
...