Is there a function $s(n)$ which tells the amount of factors any $n$ has?

Question

Is there a function $s(n)$ which tells the amount of factors any $n$ has? And if not, is it possible to write one?

Answer 1

It is called the Divisor (Counting) Function, usually denoted $\tau(n)$, $\sigma_0(n)$ or (less fancy) $d(n)$.

If $n=\prod_{k=1}^rp_k^{a_k}$ is the prime factorisation of $n$, we have $$\tau(n)=\prod_{k=1}^r(a_k+1).$$

For questions about it, see divisor-counting-function.