The Engel expansion of a positive real number $x$ is the unique non-decreasing sequence of positive integers $\{a_{1},a_{2},a_{3},\dots \} $ such that $$x=\frac{1}{a_{1}} + \frac{1}{a_{1} a_{2}} + \frac{1}{a_{1}a_{2}a_{3}} + \dots \tag{1}$$
These expansions have been studied before. Here, I propose a related notion, called the rational Engel-zeta expansion. For a positive real number $y$ it is defined as the unique non-decreasing sequence of positive integers $\{b_{1}, b_{2}, b_{3},\dots \} $ such that $$y = \frac{\zeta(2)}{b_{1}} + \frac{\zeta(3)}{b_{1}b_{2}} + \frac{\zeta(4)}{b_{1}b_{2}b_{3}} + \dots \tag{2}$$ Similarly, we can define the rational Egyptian zeta expansion for the positive real number $z$ as a non decreasing set of integers $\{c_{1},c_{2},c_{3} \dots \} \subseteq \mathbb{Z}_{>0} $ such that $$z=\frac{\zeta(2)}{c_{1}} + \frac{\zeta(3)}{c_{2}} + \frac{\zeta(4)}{c_{3}} + \dots \tag{3}$$
This is an anologue of the Egyptian fraction expansion, which has a long history. They need not be unique. For instance, the greedy algorithm gives $$\frac{5}{7} = \frac{1}{2} + \frac{1}{5} + \frac{1}{70} ,$$ whereas the continued fraction algorithm yields $$\frac{5}{7} = \frac{1}{2} + \frac{1}{6} + \frac{1}{21} .$$
Both the rational Engel-zeta and Egyptian zeta expansions are motivated by the fact that many constants can be described as a rational zeta series. For instance, we have $$S:= \sum_{n=2}^{\infty} \frac{\zeta(n)}{4^{n-1}} = \ln(8) - \frac{\pi}{2} .$$ Here, we see that $\ln(8) - \frac{\pi}{2} $ has the Engel-zeta expansion $S_{En-\zeta} = \{4,4,4,4,\dots \} $. As all Engel-zeta expansions are also Egyptian zeta expansions (but not vice versa) we may also obtain a corresponding Egyptian zeta representation: $S_{Eg-\zeta} = \{4,16,64,256,\dots \} $.
Question: have the rational Engel-zeta or Egyptian zeta fraction expansions been described in the literature before?