I'm looking for a continuous function $f(n, k)$ such that when $n, k \in \mathbb{N}$,

$f(n, k) = \begin{cases} 1, & k\mid n \\ 0, & k\nmid n \\ \end{cases}$

For example, we can define $f(n, 2) = \frac{(-1)^{n}+1}{2}$. I'm not sure how this would work for $k > 2$ however; I was thinking maybe a clever composition of sine functions, or using $e^{i \theta}$ (kind of the same thing I guess), or even using the decimal expansions of $\frac{1}{9}, \frac{1}{99},$ etc. I couldn't quite get it however, and I'd appreciate if anyone could help!


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