MathDB

N6

Part of 2018 Pan-African Shortlist

Problems(1)

2018 PAMO Shortlist: Arithmetic and geometric mean of divisors is an integer

Source: 2018 Pan-African Shortlist - N6

5/6/2019
Prove that there are infinitely many integers nn such that both the arithmetic mean of its divisors and the geometric mean of its divisors are integers.
(Recall that for kk positive real numbers, a1,a2,…,aka_1, a_2, \dotsc, a_k, the arithmetic mean is a1+a2+⋯+akk\frac{a_1 +a_2 +\dotsb +a_k}{k}, and the geometric mean is a1a2⋯akk\sqrt[k]{a_1 a_2\dotsb a_k}.)
number theoryDivisorsarithmetic meangeometric mean