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

Abstract

A positive integer $n$ is called $\phi$-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 $\phi$-practical numbers in $[1,x]$ is asymptotic to $Cx/\log x$ for some positive constant $C$ as $x\to \infty$.