On powersmooth numbers
F. F. Sharifullina · Russian Mathematics · 2017
A natural number n is called y -smooth ( y -powersmooth, respectively) for a positive number y if every prime (prime power) dividing n is bounded from above by y . Let ψ ( x, y ) and ψ *( x, y ) denote the quantity of y -smooth and y -powersmooth integers restricted by x , respectively. In this paper we investigate function ψ *( x, y ) in general. We derive formulas for finding exact calculation of ψ *( x, y ) for large x and relatively small y and give theoretical estimates for this function and for a function of the greatest powersmooth integer. This results can be used in the cryptography and number theory to estimate the convergence of factorization algorithms.