Find all functions $f : \{1, 2, ... , n,...\} \to Z$ with the following properties (i) if $a, b$ are positive integers and $a | b$, then $f(a) \ge f(b)$; (ii) if $a, b$ are positive integers then $f(ab) + f(a^2 + b^2) = f(a) + f(b)$.
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Tags: functional, functional equation, 3rd edition, algebra, number theory
Find all functions $f : \{1, 2, ... , n,...\} \to Z$ with the following properties (i) if $a, b$ are positive integers and $a | b$, then $f(a) \ge f(b)$; (ii) if $a, b$ are positive integers then $f(ab) + f(a^2 + b^2) = f(a) + f(b)$.