An estimate for the number of integers with no small factors Given integers $x > y$, I am interested in the number of positive integers  $\leq x$ and free of factors $\leq y$.  In my case $x$ and $y$ will be large. For a concrete example we can take $x=2^{100}$ and $y=2^{40}$.
Is there a computationally efficient method to estimate this number?  I was looking at Mertens' third theorem but I can't see how to use it make an efficient algorithm.
I also found On the number of positive integers ≦ x and free of prime factors > y but that is the opposite of what I want. I would like the number of positive integers $\leq x$ and free of factors $\leq y$.

The general answer to the question you ask is also related to Buchstab function.
Let $\Phi(x,y)$ be the number of positive integers $n\leq x$ which do not have any prime divisor $p< y$.
Then it is known that for any fixed $U>1$
$$
\Phi(x,y)=\frac{x w(u)}{\log y} - \frac y{\log y} +O\left(\frac x{(\log x)^2} \right)
$$
where $y=x^{\frac 1u}$ with $1\leq u\leq U$ , $y\geq 2$, and $w(u)$ satisfies
$$
w(u)=\frac 1u  \ \ \textrm{ for }1\leq u \leq 2, 
$$
$$
(uw(u))'=w(u-1) \ \ \textrm{ for }u>2.
$$
Also, it is known that $w(u)\rightarrow e^{-\gamma}$ as $u\rightarrow\infty$.
See also Montgomery & Vaughan's Multiplicative Number Theory:Classical Theory, in particular Chapter 7.
