id	title	challengeType	forumTopicId	dashedName
5900f52c1000cf542c51003e	Problem 447: Retractions C	5	302119	problem-447-retractions-c

--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);

function retractionsC() {

  return true;
}

retractionsC();

// solution required