MathDB

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

Source: 2018 Pan-African Shortlist - N6

May 6, 2019

number theoryDivisorsarithmetic meangeometric mean

Problem Statement

Prove that there are infinitely many integers

n

such that both the arithmetic mean of its divisors and the geometric mean of its divisors are integers.
(Recall that for

k

positive real numbers,

a_1, a_2, \dotsc, a_k

, the arithmetic mean is

\frac{a_1 +a_2 +\dotsb +a_k}{k}

, and the geometric mean is

\sqrt[k]{a_1 a_2\dotsb a_k}