HMMT November 2018 U34

A positive integer is called primer if it has a prime number of distinct prime factors. A positive integer is called primest if it has a primer number of distinct primer factors. A positive integer is called prime-minister if it has a primest number of distinct primest factors. Let $N$ be the smallest prime-minister number. Estimate $N$. An estimate of $E>0$ earns $\left\lfloor 20 \min \left(\frac{N}{E}, \frac{E}{N}\right)\right\rfloor$ points.