We investigate the densities of the sets of abundant numbers and of covering numbers, integers n for which there exists a distinct covering system where every modulus divides n. We establish that the set C of covering numbers possesses a natural density d (C) and prove that 0. 103230 < d (C) < 0. 103398. Our approach adapts methods developed by Behrend and Deléglise for bounding the density of abundant numbers, by introducing a function c (n) that measures how close an integer n is to being a covering number with the property that c (n) h (n) = σ (n) /n. However, computing d (C) to three decimal digits requires some new ideas to simplify the computations. As a byproduct of our methods, we obtain significantly improved bounds for d (A), the density of abundant numbers, namely 0. 247619608 < d (A) < 0. 247619658. We also show the count of primitive covering numbers up to x is O (x ( (-12 2 + ε) x x) ), which is substantially smaller than the corresponding bound for primitive abundant numbers.
McNew et al. (Wed,) studied this question.