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Totient function inside function

Source: China TST 2021, Test 2, Day 2 P4

March 22, 2021
number theorytotient functionfunction

Problem Statement

Find all functions f:Z+→Z+f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+ such that for all positive integers m,nm,n with m≥nm\ge n, f(mφ(n3))=f(m)⋅φ(n3).f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3). Here φ(n)\varphi(n) denotes the number of positive integers coprime to nn and not exceeding nn.
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