On some floor function sets

Abstract

Let $X$ be a positive integer and $t$ a real number great than 1. The family of sets $\left\{\lfloor \frac{X}{n^t}\rfloor :1\le n\le X\right\}$ have an interesting prime distribution property. We give an exact formula for the cardinality of these sets. We provide an estimate for the cardinality of the set $\left\{\lfloor \frac{X}{p}\rfloor :p\text{prime},p\le X\right\}$. For positive real $X$, we derive asymptotic formulas for the cardinality of the set $\{\lfloor f(n)\rfloor :1\le n\le X\}$ for various sets of functions.