## Retractions A

### Problem 445

Published on Saturday, 16th November 2013, 10:00 pm; Solved by 189; Difficulty rating: 50%
For every integer n>1, the family of functions f_{n,a,b} is defined
by f_{n,a,b}(`x`)≡a`x`+b mod n for a,b,`x` integer and 0<a<n, 0≤b<n, 0≤`x`<n.

We will call f_{n,a,b} a *retraction* if f_{n,a,b}(f_{n,a,b}(`x`))≡f_{n,a,b}(`x`) mod n for every 0≤`x`<n.

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

You are given that

∑ R(c) for c=C(100 000,k), and 1 ≤ k ≤99 999 ≡628701600 (mod 1 000 000 007).

(C(n,k) is the binomial coefficient).

Find ∑ R(c) for c=C(10 000 000,k), and 1 ≤k≤ 9 999 999.

Give your answer modulo 1 000 000 007.