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problem involving the tau function, Australia TST 1

Source: IMO Shortlist 2004, number theory problem 1

April 20, 2005
number theoryDivisorsequationIMO Shortlist

Problem Statement

Let τ(n)\tau(n) denote the number of positive divisors of the positive integer nn. Prove that there exist infinitely many positive integers aa such that the equation τ(an)=n \tau(an)=n does not have a positive integer solution nn.
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