Consider the set $S$ of all possible products of $n$ positive integers not exceeding $m$, that is
$S=\{ x_1x_2\cdots x_n \mid 1 \le x_1, x_2, \dots, x_n \le m \}$.
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 $F(30, 10001) \bmod 1\,000\,000\,007$.
Solution
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