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Problem 627

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Counting products

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

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乘积计数

记$S$为所有$n$个不超过$m$的正整数连乘的乘积所构成的集合，也即
$S={x_1x_2\ldots x_n|1\le x_1,x_2,\ldots,x_n\le m}.$
记$F(m,n)$为$S$中不同元素的个数。
例如，$F(9,2)=36$，而$F(30,2)=308$。

求$F(30,10001) \mod 1\ 000\ 000\ 007$。

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