## Retractions B

### Problem 446

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 $n$.

$\displaystyle F(N)=\sum_{n=1}^NR(n^4+4)$.

$F(1024)=77532377300600$.

Find $F(10^7)$.

Give your answer modulo $1\,000\,000\,007$.