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

Abstract

A positive integer n is called ϕ-practical if the polynomial X n − 1 has a divisor in Z[X] of every degree up to n. In this paper, we show that the count of ϕ-practical numbers in [1, x] is asymptotic to Cx/ log x for some positive constant C as x → ∞.

On the constant factor in several related asymptotic estimates

“…Combining the estimate (7) with Corollary 3, we obtain the following improvement of [19,Corollary 1.2].…”

“…An integer n is called ϕ-practical [18] if X n − 1 has divisors in Z[X] of every degree up to n. The name comes from the fact that X n − 1 has this property if and only if each natural number m ≤ n is a subsum of the multiset {ϕ(d) : d|n}, where ϕ is Euler's function. These numbers were first studied by Thompson [18], who showed that their counting function P ϕ (x) has order of magnitude x/ log x. Pomerance, Thompson and the author [7] established the asymptotic result…”

A Sieve Problem and Its Application

“…integers such that every can be written as a sum of distinct positive divisors of (see [4, 5, 8] and the references therein). If , then is the set of even ‐practical numbers, i.e. even integers such that the polynomial has a divisor in of every degree from to (see [2, 6, 7]).…”

“…If , then is the set of even ‐practical numbers, i.e. even integers such that the polynomial has a divisor in of every degree from to (see [2, 6, 7]).…”

“…If θ(n) = n + 1, then B is the set of even ϕ-practical numbers, i.e. even integers n such that the polynomial X n − 1 has a divisor in Z[X ] of every degree from 1 to n (see [2,6,7]). Building on earlier work by Tenenbaum [5] and Saias [4], we found [8,…”