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One of the three is the AM of other two

Source: Czech-Polish-Slovak 2012, P1

April 12, 2013
number theoryrelatively primenumber theory unsolved

Problem Statement

Given a positive integer nn, let τ(n)\tau(n) denote the number of positive divisors of nn and φ(n)\varphi(n) denote the number of positive integers not exceeding nn that are relatively prime to nn. Find all nn for which one of the three numbers n,τ(n),φ(n)n,\tau(n), \varphi(n) is the arithmetic mean of the other two.
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