The number of divisors of cube-free numbers

A cube-free number is a positive integer which is divisible by no cubic number other than 1. Given a number $n>0$, $f(n)$ is defined as follows: $$f(n)=\{\begin{array}{ll}{\sigma }_0(n)& \text{ if n is cube-free}\\ b(n)& \text{otherwise}\end{array}$$

where ${\sigma }_0(n)$ is the number of divisors of $n$, $b(n)\equiv {\sigma }_0(c(n{)}^2)$ and $c(n)$ is the maximum number $p$ such that $p^3$ divides $n$.

For example, $f(28)=6$ and $f(432)=9$.

Explanation: 28 is a cube-free number, $f(28)={\sigma }_0(28)=6$. However, 432 is not a cube-free number and 216 is the maximum cubic number which divides 432. $c(432)=6$ and $b(432)={\sigma }_0(6^2)=9$.

Let ${\displaystyle S(N)=\sum _{n=1}^{N}f(n)}$. You are given $S({10}^8)=1223779154$.

Find $S(12345678987654321)$.