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Almost multiplicative function with an iterative condition

Source: 2017 Korea Winter Program Practice Test 1 Day 2 #2

January 21, 2017
functionnumber theory

Problem Statement

Find all functions f:N→Nf : \mathbb{N} \to \mathbb{N} satisfying the following conditions:
[*]For every n∈Nn \in \mathbb{N}, f(n)(n)=nf^{(n)}(n) = n. (Here f(1)=ff^{(1)} = f and f(k)=f(k−1)∘ff^{(k)} = f^{(k-1)} \circ f.) [*]For every m,n∈Nm, n \in \mathbb{N}, ∣f(mn)−f(m)f(n)∣<2017\lvert f(mn) - f(m) f(n) \rvert < 2017.
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