Using a method due to G. J. Rieger, we show that for $1 < c < 2 $ one has, as $x$ tends to infinity $$\textrm{Card}{n \leq x : \lfloor{n^c}\rfloor} \ \textrm{ is cube-free} } = \frac{x}{\zeta(3)} + O (x^{ (c+1)/3} \log x)$$ , thus improving on a recent result by Zhang Min and Li Jinjiang.
Teerapat Srichan, 2022, Multiplicative functions of special type on Piatetski-Shapiro sequences, Mathematica Slovaca, 72, 5, pp. 1145-1150, omid:br/06504352376 doi:10.1515/ms-2022-0078.