Piatetski-Shapiro sequences

We consider various arithmetic questions for the Piatetski-Shapiro sequences $\text{fl}{n}^c$ ($n=1,2,3,\mathrm{.}\mathrm{.}\mathrm{.}$) with $c>1$, $c\in ̸\mathbb{N}$. We exhibit a positive function $\theta (c)$ with the property that the largest prime factor of $\text{fl}{n}^c$ exceeds $n^{\theta (c)-\text{eps}}$ infinitely often. For $c\in (1,\frac{149}{87})$ we show that the counting function of natural numbers $n\le x$ for which $\text{fl}{n}^c$ is squarefree satisfies the expected asymptotic formula. For $c\in (1,\frac{147}{145})$ we show that there are infinitely many Carmichael numbers composed entirely of primes of the form $p=\text{fl}{n}^c$.