Upper bound for the number of privileged words

A non-empty word w is a border of u if |w|<|u| and both prefix suffix u. privileged |u|≤1 or has that appears exactly twice in Peltomäki (2016) presented the following open problem: “Give nontrivial upper bound for B(n)”, where B(n) denotes number words length n. Let ln[0]⁡(n)=n let ln[j]⁡(n)=ln⁡(ln[j−1]⁡(n)), j,n are positive integers. We show q>1 size alphabet j≥3 an integer then there constants αj nj such thatB(n)≤αjqnln⁡nnln[j]⁡(n)∏i=2j−1ln[i]⁡(n), n≥nj. This result improves Rukavicka (2020).