Let $n$ be an integer and $S(n)$ be the set of factors of $n$.

A subset $A$ of $S(n)$ is called an antichain of $S(n)$ if $A$ contains only one element or if none of the elements of $A$ divides any of the other elements of $A$.

For example: $S(30)=\{1,2,3,5,6,10,15,30\}$.
$\{2,5,6\}$ is not an antichain of $S(30)$.
$\{2,3,5\}$ is an antichain of $S(30)$.

Let $N(n)$ be the maximum length of an antichain of $S(n)$.

Find $\sum N(n)$ for $1\le n\le {10}^8$.