Counting products

 Published on Saturday, 19th May 2018, 10:00 pm; Solved by 222;
Difficulty rating: 60%

Problem 627

Consider the set $S$ of all possible products of $n$ positive integers not exceeding $m$, that is
$S=\{ x_1x_2\dots x_n \, | \, 1 \le x_1, x_2, ..., x_n \le m \}$.
Let $F(m,n)$ be the number of the distinct elements of the set $S$.
For example, $F(9, 2) = 36$ and $F(30,2)=308$.

Find $F(30, 10001)\text{ mod }1\,000\,000\,007$.