Školjka

%V0 For positive integers $n,$ the numbers $f(n)$ are defined inductively as follows: $f(1) = 1,$ and for every positive integer $n,$ $f(n+1)$ is the greatest integer $m$ such that there is an arithmetic progression of positive integers $a_1 < a_2 < \ldots < a_m = n$ for which $$f(a_1) = f(a_2) = \ldots = f(a_m).$$ Prove that there are positive integers $a$ and $b$ such that $f(an+b) = n+2$ for every positive integer $n.$