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Prove this inequality on number of divisors of a number

Source: 2009 Jozsef Wildt International Mathematical Competition

April 26, 2020
number theoryinequalities

Problem Statement

Prove that ∑k=1n1d(k)>n+1−1\sum \limits_{k=1}^n \frac{1}{d(k)}>\sqrt{n+1}-1 For every n≥1n\geq 1, d(n)d(n) is the number of divisors of nn
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