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Problem 608

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Divisor Sums

Let $D(m,n)={\displaystyle \sum _{d|m}\sum _{k=1}^n{\sigma }_0(kd)}$ where $d$ runs through all divisors of $m$ and ${\sigma }_0(n)$ is the number of divisors of $n$.
You are given $D(3!,{10}^2)=3398$ and $D(4!,{10}^6)=268882292$.

Find $D(200!,{10}^{12})\text{ mod }({10}^9+7)$.

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因数和

记$D(m,n)={\displaystyle \sum _{d|m}\sum _{k=1}^n{\sigma }_0(kd)}$，其中$d$取遍$m$的所有因数，而${\sigma }_0(n)$表示$n$的因数数目。
已知$D(3!,{10}^2)=3398$以及$D(4!,{10}^6)=268882292$。

求$D(200!,{10}^{12})\text{ mod }({10}^9+7)$。

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