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Smooth divisors of binomial coefficients

Published on Saturday, 19th April 2014, 01:00 pm; Solved by 215;
Difficulty rating: 70%

Problem 468

An integer is called B-smooth if none of its prime factors is greater than $B$.

Let $S_B(n)$ be the largest $B$-smooth divisor of $n$.
Examples:
$S_1(10)=1$
$S_4(2100) = 12$
$S_{17}(2496144) = 5712$

Define $\displaystyle F(n)=\sum_{B=1}^n \sum_{r=0}^n S_B(\binom n r)$. Here, $\displaystyle \binom n r$ denotes the binomial coefficient.
Examples:
$F(11) = 3132$
$F(1111) \mod 1\,000\,000\,993 = 706036312$
$F(111\,111) \mod 1\,000\,000\,993 = 22156169$

Find $F(11\,111\,111) \mod 1\,000\,000\,993$.