k-free numbers in a generalized Piatetski-Shapiro sequence

¹ School of Artificial Intelligence and Big Data, Xi’an University, Xi’an, Shaanxi, China
² School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi, China

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it. ; This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 27 May 2026
Accepted: 25 June 2026

Abstract

A generalized Piatetski-Shapiro sequence is of the form

{⌊αn^c + βn + θ ⌋ } _n=1^∞ (c > 1, c ∉ ℕ)

where α, β, θ are real numbers and α > 0. The case when α = 1, β = θ = 0is the normal Piatetski-Shapiro sequence. In this article, we estimate the k-free numbers in the generalized Piatetski-Shapiro sequence, namely

A(x) = ∑ _{n ≤ x ⌊ αn^c + βn + θ ⌋ is   k-free}  1

We obtain the main term and establish bounds for the error term depending on k and c.