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Find natural function

Source: Gazeta matematică

October 28, 2019
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Problem Statement

Find the strictly monotone functions f:{0}∪N⟶{0}∪N f:\{ 0\}\cup\mathbb{N}\longrightarrow\{ 0\}\cup\mathbb{N} that satisfy the following two properties: (i)f(2n)=n+f(n), \text{(i)} f(2n)=n+f(n), for any nonnegative integers n. n. (ii)f(n) \text{(ii)} f(n) is a perfect square if and only if n n is a perfect square.
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