Eccoci.

A unit fraction contains $1$ in the numerator. The decimal representation of the unit fractions with denominators $2$ to $10$ are given:

\begin{align} 1/2 &= 0.5\\ 1/3 &=0.(3)\\ 1/4 &=0.25\\ 1/5 &= 0.2\\ 1/6 &= 0.1(6)\\ 1/7 &= 0.(142857)\\ 1/8 &= 0.125\\ 1/9 &= 0.(1)\\ 1/10 &= 0.1 \end{align}

Where $0.1(6)$ means $0.166666\cdots$, and has a $1$-digit recurring cycle. It can be seen that $1/7$ has a $6$-digit recurring cycle.

Unit fractions whose denominator has no other prime factors than $2$ and/or $5$ are not considered to have a recurring cycle.
We define the length of the recurring cycle of those unit fractions as $0$.

Let $L(n)$ denote the length of the recurring cycle of $1/n$. You are given that $\sum L(n)$ for $3 \leq n \leq 1\,000\,000$ equals $55535191115$.

Find $\sum L(n)$ for $3 \leq n \leq 100\,000\,000$.