1

Part of 2019 Mexico National Olympiad

Problems(1)

Perfect powers with divisor-function-like exponent

Source: Mexico National Olympiad 2019 Problem 1

An integer number

m\geq 1

is mexica if it's of the form

n^{d(n)}

n

is a positive integer and

d(n)

is the number of positive integers which divide

n

. Find all mexica numbers less than

2019

.
Note. The divisors of

n

12

n

d(12)=6

1, 2, 3, 4, 6, 12

are all the positive divisors of

12

.
Proposed by Cuauhtémoc Gómez

number theorynumber of divisorsPerfect powerfunction