## Retractions A

### 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 $1\,000\,000\,007$.