BALTIC-WAY 2015 Problem 16

Denote by $P(n)$ the greatest prime divisor of $n$. Find all integers $n \geq 2$ for which

$$P(n)+\lfloor\sqrt{n}\rfloor=P(n+1)+\lfloor\sqrt{n+1}\rfloor .$$

(Note: $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.)