Factor and arithmetic complexity of concatenating the $n!$

In this paper, we show that factor complexity of the infinite word $\mathfrak{F}_b$ is defined by concatenating base-$b$ representations of the $n!$ is full. Then we show that the arithmetic complexity of this word is full as well. On the other hand, $\mathfrak{F}_b$ is a disjunctive word. In number theory, this kind of words is called rich numbers.