## Smooth divisors of binomial coefficients

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