Let $S$ be the set of integers with largest prime factor bounded by a given positive integer $k$. Is there a formula for the asymptotic density of such a set $S$?



Prime factors bounded by $k$
If $k$ is fixed, then the simple bound $|S\cap[1,x]|\leq(\log\_2 x)^{\pi(k)}$ shows that the aymptotic density of $S$ is zero. For stronger bounds, see Chapter III.5 in Tenenbaum: Introduction to analytic and probabilistic number theory.


