Smooth Divisors of Binomial Coefficients

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 $F (n) = \sum_{B = 1}^{n} \sum_{r = 0}^{n} S_{B} ((\binom{n}{r}))$ . Here, $(\binom{n}{r})$ 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 $F (11 111 111) \mod 1 000 000 993$ .

Write-up coming later

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