Distribution of prime divisors of non-homogeneous Beatty sequences

Abstract

Non-homogeneous Beatty sequence is a sequence of positive integer that taking the floor value of irrational numbers. This paper using the prime counting function, $\pi(x)$ to estimate the cardinality of total distinct prime divisors of a non-homogeneous Beatty sequence. When the parameters in the non-homogeneous Beatty sequence are sufficiently large, a better estimation can be obtained. From this study, we found that for a fixed irrational $\theta>1$ and a real $\lambda>0$, the the cardinality of total distinct prime divisors is less than or equals to the prime counting function of the last term. That is $\left|A\right|\le\pi(\lfloor N\theta+\lambda\rfloor)$ where $A$ is the set of total distinct prime divisors of a non-homogeneous Beatty sequence $(\lfloor n\theta+\lambda\rfloor)$ up to $N^{\text{th}}$ term. Also, when parameters are sufficiently large, the following estimation is sharper, $\left|A\right|\le\left(\sqrt{\theta^2+\lambda}\right)\left(\log\ N\right).$