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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 
$1<c<3/2$
.
