Eccoci.

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

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

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