--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 $n$.

$F(N) = \displaystyle\sum_{n = 2}^N R(n)$.

$F({10}^7) ≡ 638\,042\,271\bmod 1\,000\,000\,007$.

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

--hints--

retractionsC() should return 530553372.

assert.strictEqual(retractionsC(), 530553372);

--seed--

--seed-contents--

function retractionsC() {

  return true;
}

retractionsC();

--solutions--

// solution required