Let $w (n)$ denote the number of different prime numbers by which $n$ is divisible. Prove that there are infinitely many natural numbers $n$ such that $w(n) < w(n + 1) < w(n + 2)$.
Source: 2015 Latvia BW TST P15
Tags: number theory, Divisors, prime divisor, prime divisors
Let $w (n)$ denote the number of different prime numbers by which $n$ is divisible. Prove that there are infinitely many natural numbers $n$ such that $w(n) < w(n + 1) < w(n + 2)$.