Almost primes of the form \([p^{1/\gamma }]\)

Let \(\mathcal {P}_r\) ${\mathcal{P}}_r$ denote an almost–prime with at most r prime factors, counted according to multiplicity. In this paper, it is proved that, for \(0.989<\gamma <1\) $0.989<\gamma <1$ , there exist infinitely many primes p such that \([p^{1/\gamma }]=\mathcal {P}_7\) $[p^{1/\gamma }]={\mathcal{P}}_7$ , which constitutes an improvement upon the previous result of Banks–Guo–Shparlinski (Indag Math (NS) 27(2):423–436, 2016) who showed that there exist infinitely many primes p such that \([p^{1/\gamma }]=\mathcal {P}_8\) $[p^{1/\gamma }]={\mathcal{P}}_8$ for \(\gamma \) $\gamma$ near to one.