For positive integers
the numbers
are defined inductively as follows:
and for every positive integer
is the greatest integer
such that there is an arithmetic progression of positive integers
for which
Prove that there are positive integers
and
such that
for every positive integer
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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 $n.$
Školjka