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

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 $\lfloor \frac{x}{n}\rfloor$, and later, a paper written by Bordellès, Dai, Heyman, and Nikolic concentrated on studying primes of the form $\lfloor \frac{x}{n^c}\rfloor$. Combining the two lines of research, in this paper we studied the smooth number of the form $\lfloor \frac{x}{n}\rfloor$ and $\lfloor \frac{x}{n^c}\rfloor$, 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.