Polynomials with divisors of every degree (Q412119)

The author is interested in those integers \(n\) for which the polynomials \(X^n-1\) has an integer polynomial divisor of every degree from 1 to \(n\). Recall that \(X^n-1\) splits into irreducible cyclotomic polynomials \(\Phi_d(X)\) of degree \(\varphi(d)\), one for each divisor \(d\) of \(n\). Thus the problem is equivalent to asking whether \(n=\sum_{d\in {\mathcal D}}\varphi(d)\) for some subset \({\mathcal D}\) of the divisors of \(n\) (such a number the author dubs `\(\varphi\)-practical', in analogy with the practical numbers for which \(n=\sum_{d\in {\mathcal D}}d\) for some subset \({\mathcal D}\) of the divisors of \(n\)). Let \(F(x)\) denote the number of \(\varphi\)-practical integers up to \(x\). The author's main result asserts that there exist two positive constants \(c_1,c_2\) such that NEWLINE\[NEWLINEc_1{x\over \log x}\leq F(x) \leq c_2{x\over \log x},NEWLINE\]NEWLINE for \(x\geq 2\). The author makes use of a similar result and its proof for the practical numbers due to \textit{E. Saias} [J. Number Theory 62, No. 1, 163--191 (1997; Zbl 0872.11039)]. In addition Stewart's Condition for practical numbers and estimates for numbers having only small prime factors (friable numbers), play a role.