# The Repeated Divisor Function and Possible Correlation with Highly Composite Numbers

## TL;DR

This paper investigates the iterative behavior of the divisor function, aiming to find the minimal number of applications needed to reach 2, and explores potential links with highly composite numbers.

## Contribution

It introduces the problem of iterating the divisor function to reach 2 and presents conjectures on its relation to highly composite numbers.

## Key findings

- Identifies the minimal iterations to reach 2 for various n
- Proposes a conjecture relating iterative divisor function to highly composite numbers
- Provides observations suggesting patterns in the divisor function's iteration

## Abstract

Let n be a non-null positive integer and $d(n)$ is the number of positive divisors of n, called the divisor function. Of course, $d(n) \leq n$. $d(n) = 1$ if and only if $n = 1$. For $n > 2$ we have $d(n) \geq 2$ and in this paper we try to find the smallest $k$ such that $d(d(...d(n)...)) = 2$ where the divisor function is applied $k$ times. At the end of the paper we make a conjecture based on some observations.