@article{Song2026, 
author = {Yanbo Song},
title = {On smooth numbers of forms        ⌊          x      n        ⌋   and        ⌊          x              n        c              ⌋},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {2},
pages = {5270-5282},
keywords = {smooth numbers, exponential sums, floor function, asymptotic formulae, largest prime factor of an integer},
url = {https://www.sciopen.com/article/10.3934/math.2026216},
doi = {10.3934/math.2026216},
abstract = {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        ⌊          x      n        ⌋  , and later, a paper written by Bordellès, Dai, Heyman, and Nikolic concentrated on studying primes of the form        ⌊        x          n      c            ⌋  . Combining the two lines of research, in this paper we studied the smooth number of the form        ⌊          x      n        ⌋   and        ⌊          x              n        c              ⌋  , where    ⌊  t  ⌋ 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.}
}