What is a possible mapping for two given sets?

Question

Suppose I a set of integers $n=\{0,1,2,...,80\}$ and $m=\{0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,...\text{repeats from start}...\}$ where $|n|=|m|=81.$ What is a function that describes the mapping $n$ to $m$? I was able to get one if $m$ were to be $\{0,1,2,3,4,5,0,1,2,....,7,8,9,10,11,12,13,14,15,...\text{repeats from start}...\}$ using a couple of floor functions and mods but that is not what I am looking for.

Answer 1

This is what you are looking for $$f(k)=\begin{cases} 2, & \left(\left\lfloor \frac{\left\lfloor \frac{k}{3}\right\rfloor }{3}\right\rfloor -2\right) \bmod 3=0 \\ 1, & \left(\left\lfloor \frac{k}{3}\right\rfloor -\left\lfloor \frac{\left\lfloor \frac{k}{3}\right\rfloor }{9}\right\rfloor \right) \bmod 2 \neq 0\\ 0, & \text{otherwise.} \end{cases}$$

Answer 2

"What is a function that maps n to m?" Um,.... what you just said was a function that maps $n$ to $m$. Functions don't need algebraic formulas or rules. Any mapping IS a function.

But if you want description:

Let $k = 27a + 9b + 3c +d$ be the unique expression of $k$ as base three numbers. That is $a,b,c,d \in \{0,1,2\}$.

$f(k) =\begin{cases}2&\text{if }b = 2\\1&\text{if }b\ne 2\text{ and }b+c\text{ is odd}\\0&b\ne 2\text{ and }b+c\text{ is even}\end{cases}$.

.....

I guess another way putting it is

$f(x) = 2$ if $[\frac{[\frac{[\frac x3]}3]}3]\equiv 2\pmod 3$

Otherwise $f(x) =1$ if $ [\frac x3]$ is odd.

Otherwise $f(x) = 0$.

where $[]$ is the greatest integer function.