Prime divisors in Beatty sequences

We study the values of arithmetic functions taken on the elements of a non-homogeneous Beatty sequence ⌊αn+β⌋⌊αn+β⌋, n=1,2,…, where α,β∈Rα,β∈R, and α>0α>0 is irrational. For example, we show that∑n⩽Nω(⌊αn+β⌋)∼NloglogNand∑n⩽N(−1)Ω(⌊αn+β⌋)=o(N), where Ω(k)Ω(k) and ω(k)ω(k) denote the number of prime divisors of an integer k≠0k≠0 counted with and without multiplicities, respectively.