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 n/k-1-Θ(√n) and n-1/k-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).