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f:N->N if prod(d|n)f(d)=2^n

Source: IMOC 2017 N1

August 13, 2021
fefunctional equationnumber theory

Problem Statement

If f:N→Rf:\mathbb N\to\mathbb R is a function such that ∏d∣nf(d)=2n\prod_{d\mid n}f(d)=2^nholds for all n∈Nn\in\mathbb N, show that ff sends N\mathbb N to N\mathbb N.
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