On the closure of the image of the generalized divisor function

For any real number s, let σs be the generalized divisor function, i.e., the arithmetic function defined by σs(n) := d | n ds , for all positive integers n. We prove that for any r > 1 the topological closure of σ−r(N+) is the union of a finite number of pairwise disjoint closed intervals I1,...,I. Moreover, for k = 1,...,, we show that the set of positive integers n such that σ−r(n) ∈ Ik has a positive rational asymptotic density dk. In fact, we provide a method to give exact closed form expressions for I1,...,I and d1,...,d, assuming to know r with sufficient precision. As an example, we show that for r = 2 it results = 3, I1 = [1, π2/9], I2 = [10/9, π2/8], I3 = [5/4, π2/6], d1 = 1/3, d2 = 1/6, and d3 = 1/2.