Is $\tau : \mathbb N \to \mathbb N$ surjective?

Question

Let $\tau :\mathbb N \to \mathbb N$ be the function where the output is the number of positive integer factors of the input.

(a) Give a specific number $n$, if possible, such that $\tau (n) = 10,017$.

(b) Is $\tau$ a surjective (onto) function?

Answer 1

b)For any $y$, we can easily find $x$ such that $\tau (x)=y$. Take $x=3^{y-1}$, then the only positive integer factors of $x$ are $3^0,3^1,3^2, \dots 3^{y-1}$. $\tau$ is surjective on a set of positive natural numbers.
a)$\tau(3^{10,016})=10,017$.