Let \(\pi_c(x)= |\{n\leq x: [n^c]\) is a prime number\(\}|\), where \(c>1\) and \([t]\) denotes the integral part of \(t\). The authors prove the inequality \(\pi_c(x)\gg x/c\log x\) for \(1< c< 243/205\approx 1.18536\) and \(x\geq x_0(c)\). They use sieve methods combined with estimates of exponential sums. To obtain the desired result, some numerical calculations are necessary which have be done with the aid of Maple.