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Maximum number of divisors

Published on Saturday, 18th October 2014, 04:00 pm; Solved by 880;
Difficulty rating: 30%

Problem 485

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

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

Find S(100 000 000,100 000).