{"app":{"locale":"en-GB"},"problem":{"content":"

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

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