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Phinding constants is Diphicult

Source: IMO SL 2020 N6

July 20, 2021
number theoryEuler s totient functionnumber of divisorsIMO ShortlistIMO Shortlist 2020

Problem Statement

For a positive integer nn, let d(n)d(n) be the number of positive divisors of nn, and let φ(n)\varphi(n) be the number of positive integers not exceeding nn which are coprime to nn. Does there exist a constant CC such that
φ(d(n))d(φ(n))≤C \frac {\varphi ( d(n))}{d(\varphi(n))}\le C for all n≥1n\ge 1
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