On smooth numbers of forms $ \left\lfloor\frac{x}{n}\right\rfloor $ and $ \left\lfloor\frac{x}{n^c}\right\rfloor $

A positive integer $ n $ is called $ y $-smooth, if its largest prime factor $ P^+(n) $ is at most $ y $. Smooth numbers are widely recognized as playing a pivotal role in analytic number theory, and the study of their distribution constitutes a core focus in relevant research. On the other hand, Bordellès, Dai, Heyman, Pan, and Shparlinski first studied the primes of the form $ \left\lfloor\frac{x}{n}\right\rfloor $, and later, a paper written by Bordellès, Dai, Heyman, and Nikolic concentrated on studying primes of the form $ \Big\lfloor\frac{x}{n^c}\Big\rfloor $. Combining the two lines of research, in this paper we studied the smooth number of the form $ \left\lfloor\frac{x}{n}\right\rfloor $ and $ \left\lfloor\frac{x}{n^c}\right\rfloor $, where $ \lfloor t\rfloor $ denoted the integral part of a real number $ t $. Clearly, studies concerning the distribution of smooth numbers play an important role in number theory, which makes our studies valuable and rather interesting.