What exactly is a Collatz-like Problem

sequences-and-series number-theory collatz-conjecture dynamical-systems

What exactly is a Collatz-like problem? Let $f:\mathbb{N} \to\mathbb{N}$. The Collatz function states that the following iterated map will eventually equal to 1:

$$f(n) = \begin{cases} n/2, & \text{if}\ 2\mid n\\ 3n+1, & \text{otherwise} \\ \end{cases}$$ I have seen many different iterated functions (1, 2, 3 )being described as Collatz-like or Collatz related once the search is for bounded or unbounded orbits. Searching for bounded or unbounded orbits is the intention of the person behind the function and doesn’t constitute a definition to me (but I could be wrong). This led me to wonder if there is a formal mathematical definition for a Collatz-like problem.


Any function of the form $3x+n$ where $n>1$ fills the bill nicely. For example:

$3x+5$ where $x=27$

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The section of this sequence that’s producing the loop is: $49,152,76,38,19,62,31,98,49$

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