Fix a positive real number $c$ and write $\gamma=1/c$. Let $\pi_c(x)$ be the number of positive integers $n\le x$ for which $\lfloor n^c\rfloor$ is prime. In many sources e.g. "Van der Corput's Method of Exponential Sums" by Graham & Kolesnik, they claim that $$\pi_c(x)=\sum_{p\le x^c}(\lfloor-p^\gamma\rfloor-\lfloor-(p+1)^\gamma\rfloor) + O(1), $$ where the sum is over primes $p\le x^c$, and $O(1)$ is some quantity bounded by a constant as $x\to\infty$.
I think this is true if $c\ge 1$ but not if $0<c<1$.
Let $p$ be a prime and $n$ be a positive integer. Indeed, we have $\lfloor n^c\rfloor=p$ if and only if $p^\gamma\le n< (p+1)^\gamma$. Therefore, $\lfloor-p^\gamma\rfloor-\lfloor-(p+1)^\gamma\rfloor$ is the number of positive integers $n$ satisfying $\lfloor n^c\rfloor=p$. Moreover, we have $n\le x$ if and only if $\lfloor n^c\rfloor\le x^c$ and $n$ does not satisfy $x^c < n^c < \lfloor x^c\rfloor + 1$.
This leads us to consider how many positive integers $n$ can satisfy $x^c < n^c < \lfloor x^c\rfloor + 1$, or equivalently, $x<n<(\lfloor x^c\rfloor + 1)^\gamma$. It appears as though the difference $$(\lfloor x^c\rfloor + 1)^\gamma-x$$ is bounded by $1$ if $c\ge 1$ and is unbounded if $0<c<1$.