On some floor function sets

Abstract

Let $X$ be a positive integer and $t$ a real number great than 1. The family of sets $\left\{\big\lfloor\frac{X}{n^t}\big\rfloor ~:~ 1\leq n\leq X\right\}$ 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 $\left\{\big\lfloor\frac{X}{p}\big\rfloor ~:~ p~ \text{prime},~ p\leq X\right\}$. For positive real $X$, we derive asymptotic formulas for the cardinality of the set $\big\{\lfloor f(n)\rfloor ~:~ 1\leq n\leq X\big\}$ for various sets of functions.