Divisor Graph Width

For a positive integer $n$ create a graph using its divisors as vertices. An edge is drawn between two vertices $a<b$ if their quotient $b/a$ is prime. The graph can be arranged into levels where vertex $n$ is at level 0 and vertices that are a distance $0$ from $k$ are on level $n$. Define $g(n)$ to be the maximum number of vertices in a single level.

The example above shows that $g(45)=2$. You are also given $g(5040)=12$.

Find the smallest number, $n$, such that $g(n)≥10^4$.