Problem

Source: 2012 Balkan Shortlist BMO N1

Tags: number theory, Sequence, Perfect Squares, consecutive, Perfect Square



A sequence $(a_n)_{n=1}^{\infty}$ of positive integers satisfies the condition $a_{n+1} = a_n +\tau (n)$ for all positive integers $n$ where $\tau (n)$ is the number of positive integer divisors of $n$. Determine whether two consecutive terms of this sequence can be perfect squares.