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2022 PUMaC Individual Finals A3

Source:

September 9, 2023
number theory

Problem Statement

Let nn be a positive integer. We call a nn-tuple (a1,...,an)(a_1, . . . , a_n) of positive integers nice if ∙\bullet gcd(a1,...,an)=1gcd (a_1, . . . , a_n) = 1, and ∙\bullet ai∣ai−1+ai+1a_i|a_{i-1} + a_{i+1}, for all i=1,...,ni = 1, . . . , n (we define a0=ana_0 = a_n and an+1=a1a_{n+1} = a1 here). Find the maximal possible value of the sum a1+...+ana_1 +...+ a_n if (a1,...,an)(a_1, . . . , a_n) is a nice nn-tuple.
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