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Problem 712

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Exponent Difference

For any integer $n>0$ and prime number $p$, define ${\nu }_p(n)$ as the greatest integer $r$ such that $p^r$ divides $n$.

Define
$$D(n,m)=\sum _{p\text{ prime}}|{\nu }_p(n)-{\nu }_p(m)|.$$
For example, $D(14,24)=4$.

Furthermore, define
$$S(N)=\sum _{1\le n,m\le N}D(n,m).$$
You are given $S(10)=210$ and $S({10}^2)=37018$.

Find $S({10}^{12})$. Give your answer modulo $1000000007$.

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指数差异

对于任意整数$n>0$和素数$p$，记${\nu }_p(n)$为使得$p^r$整除$n$的最大整数$r$。

记
$$D(n,m)=\sum _{p\text{为素数}}|{\nu }_p(n)-{\nu }_p(m)|.$$
例如，$D(14,24)=4$。

进一步地，记
$$S(N)=\sum _{1\le n,m\le N}D(n,m).$$
已知$S(10)=210$，$S({10}^2)=37018$。

求$S({10}^{12})$并将你的答案对$1000000007$取余。

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