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Perfect powers with divisor-function-like exponent

Source: Mexico National Olympiad 2019 Problem 1

November 12, 2019

number theorynumber of divisorsPerfect powerfunction

Problem Statement

An integer number

m\geq 1

is mexica if it's of the form

n^{d(n)}

, where

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

include

1

and

n

; for example,

d(12)=6

, since

1, 2, 3, 4, 6, 12

are all the positive divisors of

12

.
Proposed by Cuauhtémoc Gómez