--description--

For every integer $n > 1$, the family of functions $f_{n, a, b}$ is defined by:

$f_{n, a, b}(x) ≡ ax + b\bmod n$ for $a, b, x$ integer and $0 \lt a \lt n$, $0 \le b \lt n$, $0 \le x \lt n$.

We will call $f_{n, a, b}$ a retraction if $f_{n, a, b}(f_{n, a, b}(x)) \equiv f_{n, a, b}(x)\bmod n$ for every $0 \le x \lt n$.

Let $R(n)$ be the number of retractions for $n$.

You are given that

$$\sum_{k = 1}^{99\,999} R(\displaystyle\binom{100\,000}{k}) \equiv 628\,701\,600\bmod 1\,000\,000\,007$$

Find $$\sum_{k = 1}^{9\,999\,999} R(\displaystyle\binom{10\,000\,000}{k})$$ Give your answer modulo $1\,000\,000\,007$.

--hints--

retractionsA() should return 659104042.

assert.strictEqual(retractionsA(), 659104042);

--seed--

--seed-contents--

function retractionsA() {

  return true;
}

retractionsA();

--solutions--

// solution required