Exponent Difference

 Published on Saturday, 18th April 2020, 11:00 pm; Solved by 383;
Difficulty rating: 25%

Problem 712

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 $1\,000\,000\,007$.