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Good integer sequences

Source: China TST Test 1 Day 2 Q4

March 11, 2019
number theoryDivisibilityChina TSTLte

Problem Statement

Call a sequence of positive integers {an}\{a_n\} good if for any distinct positive integers m,nm,n, one has gcd⁡(m,n)∣am2+an2 and gcd⁡(am,an)∣m2+n2.\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2. Call a positive integer aa to be kk-good if there exists a good sequence such that ak=aa_k = a. Does there exists a kk such that there are exactly 20192019 kk-good positive integers?
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