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F(n)

Source: Cono Sur 1991-problem 6

May 29, 2006
inequalitiesnumber theoryprime numbersalgebra unsolvedalgebra

Problem Statement

Given a positive integrer number nn (n≠0n\ne 0), let f(n)f(n) be the average of all the positive divisors of nn. For example, f(3)=1+32=2f(3)=\frac{1+3}{2}=2, and f(12)=1+2+3+4+6+126=143f(12)=\frac{1+2+3+4+6+12}{6}=\frac{14}{3}. a Prove that n+12≥f(n)≥n\frac{n+1}{2} \ge f(n)\ge \sqrt{n}. b Find all nn such that f(n)=919f(n)=\frac{91}{9}.
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