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sequence of prime powers

Source: Indonesia National Math Olympiad 2021 Problem 3 (INAMO 2021/3)

November 8, 2021
number theorySequenceprimes

Problem Statement

A natural number is called a prime power if that number can be expressed as pnp^n for some prime pp and natural number nn. Determine the largest possible nn such that there exists a sequence of prime powers a1,a2,…,ana_1, a_2, \dots, a_n such that ai=ai−1+ai−2a_i = a_{i - 1} + a_{i - 2} for all 3≤i≤n3 \le i \le n.
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