CAPS 2020 Problem 2

Given a positive integer $n$ , we say that a real number $x$ is $n$ -good if there exist $n$ positive integers $a_1,...,a_n$ such that $$x=\frac{1}{a_1}+...+\frac{1}{a_n}.$$ Find all positive integers $k$ for which the following assertion is true: if $a,b$ are real numbers such that the closed interval $[a,b]$ contains infinitely many $2020$ -good numbers, then the interval $[a,b]$ contains at least one $k$ -good number.

(Josef Tkadlec, Czech Republic)