lean_workbook_plus_39881

Let $d(k)$ be the number of positive divisors of the integer $k$ . An integer $n$ is called balanced if $d(n-1) \leq d(n) \leq d(n+1) \textrm{ or } d(n-1) \geq d(n) \geq d(n+1).$ Prove there exist infinitely many balanced integers.