On some floor function sets

TL;DR

This paper investigates the properties and cardinalities of sets formed by floor functions involving division by powers and primes, providing exact formulas and asymptotic estimates for their sizes.

Contribution

It introduces exact formulas and asymptotic estimates for the cardinalities of sets defined by floor functions with divisors, including primes and powers, revealing their prime distribution properties.

Findings

Exact formula for the cardinality of floor function sets involving powers.

Asymptotic formulas for the size of sets defined by general functions.

Estimates for the cardinality of sets involving prime divisors.

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.