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Reaching a perfect square with d(n) function

Source: 2021 Turkey JBMO TST 2021 P5

May 24, 2021
number theorynumber theory proposedfunction

Problem Statement

d(n)d(n) shows the number of positive integer divisors of positive integer nn. For which positive integers nn one cannot find a positive integer kk such that d(…d(d⏟k times(n)… )\underbrace{d(\dots d(d}_{k\ \text{times}} (n) \dots ) is a perfect square.
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