Using a method due to Rieger [‘Remark on a paper of Stux concerning squarefree numbers in non-linear sequences’, Pacific J. Math. 78(1) (1978), 241–242], we prove that the Piatetski-Shapiro sequence defined by
$\{\lfloor n^c \rfloor : n\in \mathbb {N}\}$ contains infinitely many consecutive square-free integers whenever
$1<c<3/2$.