Let $\omega(n)$ be the number of distinct prime divisors of $n>1$. I could prove that: there are infinitely many positive integers $n$ such that $\omega(n+1)+\omega(n-1)>2\omega(n)$. Similarly, the reverse inequalities also hold for infinitely many positive integers.
I wish to study the case $\omega(n+1)+\omega(n-1)=2\omega(n)$. Is this equation has infinitely many solution in $\mathbb Z^+$? For example n=3.