On integers $n$ for which $X^n-1$ has a divisor of every degree

TL;DR

This paper studies integers n for which the polynomial X^n - 1 has divisors of all degrees up to n, showing that the count of such numbers up to x grows asymptotically like a constant times x divided by log x.

Contribution

It introduces the concept of φ-practical numbers and establishes their asymptotic density within the positive integers.

Findings

Number of φ-practical numbers up to x is asymptotic to C x / log x.

Provides a new classification of integers based on divisor properties of polynomial X^n - 1.

Establishes a constant C related to the distribution of φ-practical numbers.

Abstract

A positive integer $n$ is called $\varphi$-practical if the polynomial $X^n-1$ has a divisor in $\mathbb{Z}[X]$ of every degree up to $n$. In this paper, we show that the count of $\varphi$-practical numbers in $[1, x]$ is asymptotic to $C x/\log x$ for some positive constant $C$ as $x \rightarrow \infty$.