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a) s(n) = n/2 (n + 1- \varphi (n)) , b) s (n) = s (n + 2021) no solutions

Source: VMO 2021 P4 Vietnam National Olympiad

December 26, 2020
number theorySumphi function

Problem Statement

For an integer n≥2 n \geq 2 , let s(n) s (n) be the sum of positive integers not exceeding n n and not relatively prime to n n . a) Prove that s(n)=n2(n+1−φ(n)) s (n) = \dfrac {n} {2} \left (n + 1- \varphi (n) \right) , where φ(n) \varphi (n) is the number of integers positive cannot exceed n n and are relatively prime to n n . b) Prove that there is no integer n≥2 n \geq 2 such that s(n)=s(n+2021) s (n) = s (n + 2021)
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