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What is the mean of exponents of all naturals?

Source: Brazilian Undergrad MO 2022 Problem 4

May 12, 2022

real analysislimitsmeanExponentsBrazilian Undergrad MO 2021

Problem Statement

For every positive integeer

n>1

, let

k(n)

the largest positive integer

k

such that there exists a positive integer

m

such that

n = m^k

.
Find

lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n}