Cube-full Divisors

A positive integer $n$ is considered cube-full, if for every prime $p$ that divides $n$, so does $p^3$. Note that $1$ is considered cube-full.

Let $s(n)$ be the function that counts the number of cube-full divisors of $n$. For example, $1$,  $8$ and $16$ are the three cube-full divisors of $16$. Therefore, $s(16) = 3$.

Let $S(n)$ represent the summatory function of $s(n)$, that is $S(n) = \sum_{i=1}^n s(i)$.

You are given $S(16)=19$, $S(100)=126$ and $S(10000)=13344$.

Find $S(10^{18})$.