The Problem

An Egyptian fraction is the sum of reciprocals of distinct positive integers. For example
1312=12+13+14

For any number n, define f(n) as the smallest positive difference between two Egyptian fractions (1/a1 + 1/a2 + 1/a3 …) – (1/b1 + 1/b2 + 1/b3 …), where all the a’s and b’s are distinct numbers ≤ n.

Find the smallest n satisfying f(n) < 1/1013.

A Simple Example

n=10
There are 59049 ways to pick two disjoint sets of numbers from {1, 2,.. 10}. Each set corresponds to a fraction by summing its reciprocals. We ignore the fractions with zero or negative difference, e.g. (1/2) – (1/3 + 1/6) = 0.
From those remaining, the smallest difference occurs at (1/7 + 1/8) – (1/6 + 1/10) = 1/840
So f(10) = 1/840