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Romania EGMO TST 2022 Day 1 P4

Source:

February 15, 2022
romaniaEGMOnumber theoryfunction

Problem Statement

For every positive integer N≥2N\geq 2 with prime factorisation N=p1a1p2a2⋯pkakN=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k} we define f(N):=1+p1a1+p2a2+⋯+pkak.f(N):=1+p_1a_1+p_2a_2+\cdots+p_ka_k. Let x0≥2x_0\geq 2 be a positive integer. We define the sequence xn+1=f(xn)x_{n+1}=f(x_n) for all n≥0.n\geq 0. Prove that this sequence is eventually periodic and determine its fundamental period.
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