A Note on the Maximum Number of k-Powers in a Finite Word

A power is a concatenation of k copies of a word u, for a positive integer k; the power is also called a k-power and k is its exponent. We prove that for any k 2, the maximum number of different non-empty k-power factors in a word of length n is between nk-1- (n) and n-1k-1. We also show that the maximum number of different non-empty power factors of exponent at least 2 in a length-n word is at most n-1. Both upper bounds generalize the recent upper bound of n-1 on the maximum number of different square factors in a length-n word by Brlek and Li (2022).