Variations on a question concerning the degrees of divisors of \(x^{n}-1\) (Q2454453)

Let \(f(x)= x^n- 1\) for \(n\in\mathbb N\). The integer \(n\) is said to be \(\varphi\)-practical if \(f(x)\) has a divisor in \(\mathbb Z[x]\) of every degree up to \(n\). If \(f(x)\) has the corresponding property in the ring \(\mathbb F_p[x]\) for \(p\) a rational prime, then \(n\) is called \(p\)-practical, and if \(n\) is \(p\)-practical for every prime \(p\), then \(n\) is said to be \(\lambda\)-practical. Denote by \(F_\varphi(X)\), \(F_\lambda(X)\), \(F_p(X)\) the number of integers \(n\leq X\) that are \(\varphi\)-practical, \(\lambda\)-practical, \(p\)-practical, respectively. The aim of this paper is to compare the number of \(n\leq X\) in these three categories. In [J. Number Theory 132, No. 5, 1038--1053 (2012; Zbl 1287.11113)], the author established that there exist positive constants \(c_1\), \(c_2\) such that \[ c_1{X\over\log X}\leq F_\varphi(X)\leq c_2{X\over\log X}.\tag{\(*\)} \] More recently in [Acta Arith. 175, No. 3, 225--243 (2016; Zbl 1364.11143 (2016; Zbl 1364.11143)], the author et al. replaced the bounds in \((*)\) by an asymptotic formula. Bounds for \(F_\lambda(X)\) analogous to those in \((*)\) are obtained in Proposition 5.1 of this paper. In Theorem 1.1 it is shown that the order of magnitude of the number of \(n\leq X\) that are \(\lambda\)-practical but not \(\varphi\)-practical is \({X\over\log X}\). In Theorem 1.2 it is established that the number of \(n\leq X\) that are \(p\)-practical but not \(\lambda\)-practical is \(\gg{X\over\log X}\); however the precise order of magnitude of this set is not yet known, for the true order of magnitude of \(F_p(X)\) is so far unknown. The proofs depend on alternative characterizations of \(\varphi\)-practical and \(\lambda\)-practical numbers, and also on properties of other special sequences that have previously been investigated in the literature.