Let $S(n)$ denote the number of $x$ such that $1 \leq x < n$ and $x^{23} \equiv 1 \pmod{n}$.

For example, $S(4) = 1$ because $1^{23} \equiv 1 \pmod{4}$, but $2^{23} \equiv 0 \pmod{4}$ and $3^{23} \equiv 3 \pmod{4}$.

Let $F(k)$ denote the number of $n$ between $1$ and $10^{18}$ such that $S(n) = k$.

Find the summation $\sum_{n=10^9}^{\infty} F(n) \mod (10^9 + 7)$.