Almost primes of the form [p1/γ]

Let Pᵣ denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, it is proved that, for 0.989<γ<1, there exist infinitely many primes p such that [p1/γ]=P₇, which constitutes an improvement upon the previous result of Banks-Guo-Shparlinski [4] who showed that there exist infinitely many primes p such that [p1/γ]=P₈ for γ near to one.