id	title	challengeType	forumTopicId	dashedName
5900f52a1000cf542c51003c	Problem 445: Retractions A	5	302117	problem-445-retractions-a

--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);

function retractionsA() {

  return true;
}

retractionsA();

// solution required