It is well-known that any rational number can be represented using a sum of distinct Egyptian fractions (that is, rational fractions of the form $1/n$ with $n\in\mathbb{N}$). This may be proven by establishing a greedy algorithm that constructs a sequence of such decompositions. For instance, \begin{align} 1 &= \frac12+\frac13+\frac16 \\ &= \frac12+\frac13+\frac17+\frac1{42}\\ &= \frac12+\frac13+\frac17+\frac1{43}+\frac1{1806}\\ &= \cdots\\&=\sum_{k=1}^\infty \frac{1}{x_k} \tag{1} \end{align} where $x_1=2$ and $x_k = 1+\prod_{j=1}^{k-1}x_j$ for $k>1$ (i.e. Sylvester's sequence A000058).

As noted by Bergner & Walker [1], such Egyptian fractions are relevant in the context of groupoid cardinality. To briefly review: Given a groupoid $G$, the groupoid cardinality is defined [2] by $|G|=\sum_{[\bullet]\in G}\frac{1}{\#\text{Aut}(\bullet)}$ where $[\bullet]$ is a component of $G$ and $\#\text{Aut}(\bullet)$ is the order of the automorphism group of $\bullet$. If $G$ is a group, then this reduces to $|G| = \frac{1}{\# G}$, and if we form the groupoid $G\coprod H$ as a direct union of groups $G,H$ we have $\textstyle |G\coprod H| =\frac{1}{\#G}+\frac{1}{\#H}$.

Here the link to Egyptian fractions arises: Since every rational number can be represented in terms of Egyptian fractions, so too can every rational fraction be obtained as the cardinality of some groupoid. As a key example, since $|\mathbb{Z}/n|=1/n$ the representations in $(1)$ imply $$1 = |\mathbb{Z}/2\coprod \mathbb{Z}/2|=|\mathbb{Z}/2\coprod \mathbb{Z}/3\coprod \mathbb{Z}/6|=\cdots.$$

But this construction, while valid, is quite artificial. In particular, it is far from obvious to me why each successive groupoid has unit cardinality. This suggests the following question: **Is there a natural sequence of groupoids $\{G_k\}$, each of unit cardinality, which generate $(1)$?**

References:

[1] John Baez, James Dolan, "From Finite Sets to Feynman Diagrams" (arXiv)

[2] Julia Bergner, Christopher Walker, "Groupoid Cardinality and Egyptian Fractions" (JSTOR)