Concerning types of square-free numbers and comparing sizes of their subsets.

Question

Call a square-free a 2-prime if it has exactly two prime divisors. Call a square-free a 3-prime if it has exactly three prime divisors,etc. Does there exist an integer sufficiently large N such that the number of 3-primes less than N is two times (or more) as large as the number of 2-primes less than N ?

Answer 1

There are about $\frac{x\log\log x}{\log x}$ 2-primes up to $x$ and about $\frac{x(\log\log x)^2}{2\log x}$ 3-primes up to $x$, so the crossover should happen roughly when $\log\log x=4$ which is $e^{e^4}\approx2^{78.8}.$