  We consider various arithmetic questions for the Piatetski-Shapiro sequences
$\fl{n^c}$ ($n=1,2,3,...$) with $c>1$, $c\not\in\N$. We exhibit a positive
function $\theta(c)$ with the property that the largest prime factor of
$\fl{n^c}$ exceeds $n^{\theta(c)-\eps}$ infinitely often. For
$c\in(1,\tfrac{149}{87})$ we show that the counting function of natural numbers
$n\le x$ for which $\fl{n^c}$ is squarefree satisfies the expected asymptotic
formula. For $c\in(1,\tfrac{147}{145})$ we show that there are infinitely many
Carmichael numbers composed entirely of primes of the form $p=\fl{n^c}$.
