On integers \(n\) for which \(X^n-1\) has a divisor of every degree (Q2833597)

Various authors, including Erdős, have studied practical numbers \(n\), defined to be positive integers such that each \(m\) \((1\leq m\leq n)\) can be expressed as a subsum of the divisors of \(n\). An asymptotic formula for the number of such integers up to \(x\) was obtained by the third author [Q. J. Math. 66, No. 2, 743--758 (2015; Zbl 1338.11087)]. A positive integer \(n\) is said to be \(\varphi\)-practical if the polynomial \(x^n-1\) has divisors of every degree up to \(n\). Since \(n=\sum_{d|n}\varphi(d)\), this means that every integer \(m\) with \(1\leq m\leq n\) is a subsum of the multiset \(\{\varphi(d): d|n\}\). Let \(P_\varphi(x)\) denote the number of \(\varphi\)-practical numbers up to \(x\). The second author [J. Number Theory 132, No. 5, 1038--1053 (2012; Zbl 1287.11113)] proved that there are positive numbers \(c_1\), \(c_2\) such that NEWLINE\[NEWLINEc_1{x\over\log x}\leq P_\varphi(x)\leq c_2{x\over\log x}.NEWLINE\]NEWLINE The aim of this interesting paper is to establish that there exists a positive number \(C\) such that NEWLINE\[NEWLINEP_\varphi(x)= C{x\over\log x}\Biggl(1+ O\Biggl({1\over\log x}\Biggr)\Biggr).\tag{\(*\)}NEWLINE\]NEWLINE There are several parts to the proof and these benefit from ideas developed in the papers quoted above. An early step is to count the number \(B_m(x)\) of integers \(n= mb\leq x\) with arbitrary fixed \(m\) and squarefree \(b\) satisfying certain conditions. It is known that \(B_1(x)\) equals the number of squarefree \(\varphi\)-practical numbers up to \(x\). The proof continues by showing that each \(\varphi\)-practical number \(n\) has a unique starter \(m\) such that \(m|n\) and \({n\over m}\) is squarefree together with some other conditions. Then \((*)\) is established by summing \(B_m(x)\) over starters \(m\) and verifying that starters \(m\) are sufficiently scarce for the convergence of certain series whose terms occur in the formula for \(B_m(x)\).