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 kn-1 n - 1 - Theta(root n) root n ) andn-1 n - 1 k-1 - 1 . We also show that the maximum number of different non-empty power factors of exponent at least 2 in a length-n 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 n word by Brlek and Li (2022).