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Prefix sums of divisors are perfect squares

Source: ISL 2021 N3

July 12, 2022
number theoryIMO Shortlist

Problem Statement

Find all positive integers nn with the following property: the kk positive divisors of nn have a permutation (d1,d2,…,dk)(d_1,d_2,\ldots,d_k) such that for i=1,2,…,ki=1,2,\ldots,k, the number d1+d2+⋯+did_1+d_2+\cdots+d_i is a perfect square.
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