On integers $n$ for which $X^n-1$ has a divisor of every degree
Volume 175 / 2016
Acta Arithmetica 175 (2016), 225-243
MSC: Primary 11N25; Secondary 11N37.
DOI: 10.4064/aa8354-6-2016
Published online: 15 September 2016
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 $n$. 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 $x \rightarrow \infty$.
