Piatetski-Shapiro sequences
Roger C. Baker, William D. Banks, Jörg Brüdern, Igor E. Shparlinski, Andreas J. Weingartner
Abstract
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
θ(c)
with the property that the largest prime factor of $\fl{n^c}$ exceeds $n^{\theta(c)-\eps}$ infinitely often. For
c∈(1,
149
87
)
we show that the counting function of natural numbers
n≤x
for which $\fl{n^c}$ is squarefree satisfies the expected asymptotic formula. For
c∈(1,
147
145
)
we show that there are infinitely many Carmichael numbers composed entirely of primes of the form $p=\fl{n^c}$.