SILK-ROAD 2016 Problem 3

Given natural numbers $a,b$ and function $f: \mathbb{N} \to \mathbb{N}$ such that for any natural number $n, f\left( n+a \right)$ is divided by $f\left( {\left[ {\sqrt n } \right] + b} \right)$ . Prove that for any natural $n$ exist $n$ pairwise distinct and pairwise relatively prime natural numbers ${{a}_{1}}$ , ${{a}_{2}}$ , $\ldots$ , ${{a}_{n}}$ such that the number $f\left( {{a}_{i+1}} \right)$ is divided by $f\left( {{a}_{i}} \right)$ for each $i=1,2, \dots ,n-1$ .

(Here $[x]$ is the integer part of number $x$ , that is, the largest integer not exceeding $x$ .)