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Prove this inequality on perfect number

Source: 2009 Jozsef Wildt International Math Competition

April 27, 2020
number theoryinequalitiesPerfect Numbers

Problem Statement

Let aa, nn be positive integers such that ana^n is a perfect number. Prove that anμ>μ2a^{\frac{n}{\mu}}> \frac{\mu}{2} where μ\mu denotes the number of distinct prime divisors of ana^n
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