Approximating number of integers with smallest prime factor above a given number?

Question

I am trying to approximate the number of integers less than $x$ that have no prime factors less than $B$. Is such an approximation known? If so, how good is it?

Note that this should not be confused with approximating $x - \Psi(x, B)$, since that would approximate the numbers less than $x$ that have at least one prime factor greater than or equal to $B$.

Answer 1

Hint: How many primes are there in between B and X ? How many of their products are less then X ? For the first question, see prime-counting function $\pi(n)\simeq\dfrac n{\ln n}$ . For the second, $p_k^2\in[B^2,x]$ would be a necessary, though insufficient, first condition.