We sharpen a 1980 theorem of Erdős and Wagstaff on the distribution of positive integers having a large shifted prime divisor. Specifically, we obtain precise estimates for the quantity N (x, y) := #{n ≤ x : − 1 | n for some − 1 > y, prime}, in essentially the full range of x and y. We then present an application to a problem in arithmetic statistics. Let T CM (d) denote the largest order of a torsion subgroup of a CM elliptic curve defined over a degree d number field. Recently, Bourdon, Clark, and Pollack showed that the set of d with T CM (d) > y has upper density tending to 0, as y → ∞. We quantify the rate of decay to 0, proving that the upper and lower densities of this set both have the form (log y) −β+o(1) , where β = 1 − 1+log log 2 log 2 (the Erdős-Ford-Tenenbaum constant).