Problem

Source: IV International Festival of Young Mathematicians Sozopol 2013, Theme for 10-12 grade

Tags: number theory, Sequence



Determine all increasing sequences $\{a_n\}_{n=1}^\infty$ of natural numbers with the following property: for each two natural numbers $i$ and $j$ (not necessarily different), the numbers $i+j$ and $a_i+a_j$ have an equal number of distinct natural divisors.