On power smooth numbers

A natural number $n$ is called the $y$-power smooth for some positive number $y$ if every prime power dividing $n$ is bounded from above by the number $y$. Let us denote by $\psi^*(x,y)$ the amount of $y$-power smooth integers in the range from $0$ to $x$. In this paper we investigate the function $\psi^*(x,y)$ and $y$-power smooth numbers in general. We derive formulas for finding exact calculation of $\psi^*(x,y)$ for large $x$ and relatively small $y$, and give theoretical estimates for this function and for function of the greatest powersmooth number. This results can be used in the cryptography and number theory to estimate the convergence of the factorization algorithms.