A Function in the Number Theory

TL;DR

This paper introduces a function in number theory that determines the smallest integer whose factorial is divisible by a given number, utilizing prime-based functions for calculation.

Contribution

It defines a new number-theoretic function and provides a method to compute it using prime-related functions in a power base.

Findings

Defined the function ta(n) for divisibility by factorials.

Established a prime-based approach to compute ta(n).

Potential applications in divisibility and factorial-related problems.

Abstract

In this paper one constructs a function $\eta$ with the property that if $n$ is non-null then $\eta(n)$ is the smallest integer such that $\eta(n)!$ is divisible by $n$. In order to calculate it one considers, for each prime $p$, the associated function $\eta_{p}(n)$ in a power base.