Let r≥ 2 be a fixed integer. A positive integer n is called r-free if in the canonical representation of n into prime powers each exponent is < r. The integer 1 is considered to be r-free. In this paper, we consider Q<inf>r</inf>(x; α, β) , which is the number of r-free integers of Beatty sequence ⌊ αn+ β⌋ , 1 ≤ n≤ x, for α> 1 irrational and with bounded partial quotients, β∈ [0 , α). We prove that, as x→ ∞Qr(x;α,β)=xζ(r)+O(x(r+1)/2rlog3x),which improves Victorovich’s result in the case of square-free integers. Moreover, we also prove there exist infinitely many consecutive square-free numbers of the forms ⌊ αn+ β⌋ , ⌊ αn+ β⌋ + 1 , which improves Dimitrov’s result in 2019.
