Piatetski-Shapiro sequences

TL;DR

This paper investigates the arithmetic properties of Piatetski-Shapiro sequences, establishing results on prime factors, squarefreeness, and Carmichael numbers for sequences defined by non-integer powers.

Contribution

It provides new bounds on prime factors, asymptotic formulas for squarefree counts, and existence of Carmichael numbers within Piatetski-Shapiro sequences.

Findings

Largest prime factor exceeds $n^{\theta(c)-\eps}$ infinitely often.

Asymptotic formula for count of squarefree numbers in the sequence.

Existence of infinitely many Carmichael numbers with primes of the form $\fl{n^c}$.

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 $\theta(c)$ 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 $p=\fl{n^c}$.