## Retractions C

Published on Saturday, 16th November 2013, 10:00 pm; Solved by 219;
Difficulty rating: 95%

### Problem 447

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 $n$.

$\displaystyle F(N)=\sum_{n=2}^N R(n)$.
$F(10^7)\equiv 638042271 \mod 1\,000\,000\,007$.

Find $F(10^{14})$.
Give your answer modulo $1\,000\,000\,007$.