## Retractions A

Published on Saturday, 16th November 2013, 10:00 pm; Solved by 292;
Difficulty rating: 50%

### Problem 445

For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined by
$f_{n,a,b}(x)\equiv a x + b \mod n\,\,\,$ for $a,b,x$ integer and $0< a <n, 0 \le b < n,0 \le x < n$.

We will call $f_{n,a,b}$ a retraction if $\,\,\, f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \mod n \,\,\,$ for every $0 \le x < n$.
Let $R(n)$ be the number of retractions for $n$.

You are given that
$\displaystyle \sum_{k=1}^{99\,999} R(\binom {100\,000} k) \equiv 628701600 \mod 1\,000\,000\,007$.

Find $\displaystyle \sum_{k=1}^{9\,999\,999} R(\binom {10\,000\,000} k)$.
Give your answer modulo $1\,000\,000\,007$.