Is this a good function that defines collatz

Question

I am trying to find a function that perfectly defines the first iteration of the Collatz Conjecture. Is this good enough? Or is there a better and more compact function?

$f\left(x\right)=\left(\frac{x}{2}-\operatorname{floor}\left(\frac{x}{2}\right)\right)\cdot2\cdot\left(3x+1\right)+\left(1-\left(\left(\frac{x}{2}-\operatorname{floor}\left(\frac{x}{2}\right)\right)\cdot2\right)\right)\cdot\frac{x}{2}$

You can replace "floor" by "rfloor ($\lfloor$)" and "lfloor ($\rfloor$)". — Dominique, Dec 18 '24 at 08:56
Floor functions are not "elementary" in any useful sense, see e.g. n! using floors. Rewriting the function in an algebraic one-liner will not be useful. — Joshua P. Swanson, Dec 18 '24 at 09:09
Probably easier to use $(-1)^n$ to determine the parity of $n$, or $\cos n\pi$. — Chris Lewis, Dec 18 '24 at 09:26

score 7 · Answer 1 · answered Dec 18 '24 at 08:51

7

There is a better way:

$$f(x)=\begin{cases}3x+1 & \text{if } x \text{ is odd}\\ \frac x2 & \text{if } x \text{ is even}\end{cases}$$

Unlike your version, if a human is reading it, this second version is much easier to understand.

answered Dec 18 '24 at 08:51

5xum

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score 4 · Answer 2 · answered Dec 18 '24 at 09:43

4

If you really want one function keep it simple

$$f(x)=(3x+1)\cdot \frac{1-(-1)^{x}}{2} + \frac{x}{2}\cdot \frac{1+(-1)^{x}}{2}$$ but again, I would personally discourage its use for the same reasons as @5xum's

answered Dec 18 '24 at 09:43

b00n heT

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Is this a good function that defines collatz

2 Answers2