Problem

Source: Romanian JBMO TST 1998, Day 1, P3

Tags: number theory



Let $ n $ be a natural number. Find all integer numbers that can be written as $$ \frac{1}{a_1} +\frac{2}{a_2} +\cdots +\frac{n}{a_n} , $$where $ a_1,a_2,...,a_n $ are natural numbers.