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In number theory, completely additive arithmetic functions are arithmetic functions $\scriptstyle a(n),\, n \,\in\, \N^+, \,$ such that

$a(mn) = a(m) + a(n),\quad m,\, n \in \N^+. \,$

Obviously, $\scriptstyle a(1)$ must be 0. An example is Ω(n), the number of prime factors of n.