Cube-full Divisors

 Published on Saturday, 21st December 2019, 07:00 pm; Solved by 721;
Difficulty rating: 15%

Problem 694

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 $S(n)=\displaystyle\sum_{i=1}^n s(i)$.

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

Find $S(10^{18})$.