Let $N(i)$ be the smallest integer $n$ such that $n!$ is divisible by $(i!)^{1234567890}$
Let $S(u)=\sum N(i)$ for $10 \le i \le u$.
$S(1000)=614538266565663$.
Find $S(1\,000\,000) \bmod 10^{18}$.
Solution
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