  Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors,
counted according to multiplicity. In this paper, it is proved that, for
$0.989<\gamma<1$, there exist infinitely many primes $p$ such that
$[p^{1/\gamma}]=\mathcal{P}_7$, which constitutes an improvement upon the
previous result of Banks-Guo-Shparlinski [4] who showed that there exist
infinitely many primes $p$ such that $[p^{1/\gamma}]=\mathcal{P}_8$ for
$\gamma$ near to one.
