Number of prime divisors

Question

Is there a way to express all the prime divisors of a natural number x as a function? Thanks in advance.

Answer 1

As Daniel Hast points out with this link to mathworld, the function that inputs an integer $x$ and outputs the number of distinct prime divisors of $x$ is usually denoted $\omega(x)$.

You can write its definition in a variety of ways. $$\begin{align*} \omega(x)&=\sum_{\large\substack{p\,\mid\, x\\p\text{ prime}}}1\\\\ \omega(x)&=\#\{p\in\mathbb{N}:p\text{ is prime, and }p\mid x\}\\\\ &\text{etc.} \end{align*}$$ However, it is just as valid and rigorous to write the definition simply as

$\omega(x)$ is defined to be the number of distinct prime divisors of $x$.

This is also "expressing $\omega$ as a function".