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Problem 712


Problem 712


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
$$D(n, m) = \sum_{p \text{ prime}} \left| \nu_p(n) - \nu_p(m)\right|.$$
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 $1\ 000\ 000\ 007$.


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$r$。


$$D(n, m) = \sum_{p \text{为素数}} \left| \nu_p(n) - \nu_p(m)\right|.$$
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$1\ 000\ 000\ 007$取余。