MathNet 2022-part-02 — 197 problems (EN–VI) Question 182

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1, d_2, \dots, d_k)$ such that for every $i = 1, 2, \dots, k$, the number $d_1 + d_2 + \dots + d_i$ is a perfect square.