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Arithmetic mean of divisors and of coprime numbers

Source: Dutch TST 2024, 1.1

June 28, 2024
number theorynumber theory proposedDivisorscoprime

Problem Statement

For a positive integer nn, let α(n)\alpha(n) be the arithmetic mean of the divisors of nn, and let β(n)\beta(n) be the arithmetic mean of the numbers k≤nk \le n with gcd(k,n)=1\text{gcd}(k,n)=1. Determine all positive integers nn with α(n)=β(n)\alpha(n)=\beta(n).
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