{"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 $n$.

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

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