Retractions C

For every integer n>1, the family of functions f_n,a,b is defined by f_n,a,b(x)≡ax+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.

F(N)=∑R(n) for 2≤n≤N. F(10⁷)≡638042271 (mod 1 000 000 007).

Find F(10¹⁴) (mod 1 000 000 007).

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