Factorials Divisible by a Huge Integer

Let $N (i)$ be the smallest integer $n$ such that $n!$ is divisible by $(i!)^{1234567890}$

Let $S (u) = \sum N (i)$ for $10 \leq i \leq u$ .

$S (1000) = 614538266565663$ .

Find $S (1 000 000) mod 10^{18}$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.