  Let $\pi(x;\gamma_1,\gamma_2)$ denote the number of primes $p$ with
$p\leqslant x$ and $p=\lfloor n^{1/\gamma_1}_1\rfloor=\lfloor
n^{1/\gamma_2}_2\rfloor$, where $\lfloor t\rfloor$ denotes the integer part of
$t\in\mathbb{R}$ and $1/2<\gamma_2<\gamma_1<1$ are fixed constants. In this
paper, we show that $\pi(x;\gamma_1,\gamma_2)$ holds an asymptotic formula for
$21/11<\gamma_1+\gamma_2<2$, which constitutes an improvement upon the previous
result of Baker [1].
