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8

Part of 2019 Switzerland - Final Round

Problems(1)

Resistant integers

Source: Switzerland 2019, final round, problem 8

3/4/2019
An integer n≥2n\ge2 is called resistant, if it is coprime to the sum of all its divisors (including 11 and nn). Determine the maximum number of consecutive resistant numbers.
For instance: * n=5n=5 has sum of divisors S=6S=6 and hence is resistant. * n=6n=6 has sum of divisors S=12S=12 and hence is not resistant. * n=8n=8 has sum of divisors S=15S=15 and hence is resistant. * n=18n=18 has sum of divisors S=39S=39 and hence is not resistant.
number theoryDivisibilitysum of divisors