On the distribution of numbers related to the divisors of xn − 1

Let n1,⋯,nrn1,⋯,nr be any finite sequence of integers and let S be the set of all natural numbers n   for which there exists a divisor d(x)=1+∑i=1deg(d)cixi of xn−1xn−1 such that ci=nici=ni for 1≤i≤r1≤i≤r. In this paper we show that the set S has a natural density. Furthermore, we find the value of the natural density of S.