How many integers up to 200 have exactly four factors?

Question

Is there an easier way to work this out without actually counting the amount of numbers?

I have started to count and have realised that nearly only numbers which are only divisible only by 1,2, half of n, and n have for factors.

Answer 1

Hint: If $n=p_1^{e_1}\cdots p_m^{e_m}$, then the number of divisors of $n$ is exactly $(e_1+1)\cdots(e_m+1) \ge 2^m$.

Or, if you want an ad hoc solution, consider this:

If $n$ has three prime factors, then it has more than four divisors.