  Let $N(x,y)$ denote the number of integers $n\le x$ which are divisible by a
shifted prime $p-1$ with $p>y$, $p$ prime. Improving upon recent bounds of
McNew, Pollack and Pomerance, we establish the exact order of growth of
$N(x,y)$ for all $x\ge 2y\ge 4$.
