Maximum Number of Divisors

Let $d (n)$ be the number of divisors of $n$ .
Let $M (n, k)$ be the maximum value of $d (j)$ for $n \leq j \leq n + k - 1$ .
Let $S (u, k)$ be the sum of $M (n, k)$ for $1 \leq n \leq u - k + 1$ .

You are given that $S (1000, 10) = 17176$ .

Find $S (100 000 000, 100 000)$ .

Write-up coming later

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