Let n be a random integer (sampled from {1,..,X} for some large

X). It is a classical fact that, typically, n will have around (log n)^{log

2} divisors. Must some of these be close together? Hooley's Delta function

Delta(n) is the maximum, over all dyadic intervals I = [t,2t], of the

number of divisors of n in I. I will report on joint work with Kevin Ford

and Dimitris Koukoulopoulos where we conjecture that typically Delta(n) is

about (log log n)^c for some c = 0.353.... given by an equation involving

an exotic recurrence relation, and then prove (in some sense) half of this

conjecture, establishing that Delta(n) is at least this big almost surely.

For the most part I will discuss a model combinatorial problem about

representing integers in many ways as sums of elements from a random set.