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

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 $x \rightarrow \infty$.