Let 1= d 1( n)< d 2( n)<⋯< d τ( n) = n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are y-dense iff max 1⩽ i< τ( n) d i+1 ( n)/ d i ( n)⩽ y. Let D( x, y, z) be the number of positive integers not exceeding x whose divisors are y-dense and whose prime divisors are bigger than z, and let u= log x/ log y , and v= log x/ log z . We show that x −1D(x,y,z) log z is equivalent, in a large region, to a function d( u, v) which satisfies a difference-differential equation. Using that equation we find that d( u, v)≍(1− u/ v)/( u+1) for v⩾3+ ε. Finally, we show that d( u, v)= e − γ d( u)+ O(1/ v), where γ is Euler's constant and d( u)∼ x −1 D( x, y,1), for fixed u. This leads to a new estimate for d( u).