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Functions where d(f(x)) = x

Source: 2012 Indonesia Round 2 TST 2 Problem 4

March 4, 2012
functionnumber theory unsolvednumber theory

Problem Statement

Let N\mathbb{N} be the set of positive integers. For every n∈Nn \in \mathbb{N}, define d(n)d(n) as the number of positive divisors of nn. Find all functions f:N→Nf : \mathbb{N} \rightarrow \mathbb{N} such that: a) d(f(x))=xd(f(x)) = x for all x∈Nx \in \mathbb{N} b) f(xy)f(xy) divides (x−1)yxy−1f(x)(x-1)y^{xy-1}f(x) for all x,y∈Nx,y \in \mathbb{N}
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