Let $u_1=1$ and $u_{n+1}=u_n(u_n+1)$, so that $\sum_{k\geq 1}\frac{1}{u_k+1}$ and $u_k=\lfloor c_0^{2^k}+1\rfloor$ for $k\geq 1$, where\[c_0=\lim u_n^{1/2^n}=1.264085\cdots.\]Let $a_1<a_2<\cdots $ be any other sequence with $\sum \frac{1}{a_k}=1$. Is it true that\[\liminf a_n^{1/2^n}<c_0=1.264085\cdots?\]
An earlier interpretation of this question on this site defined $u_1=2$ and $u_{n+1}=u_n^2-u_n+1$ (
Sylvester's sequence), which is the same sequence shifted by $1$; we use the phrasing above as more faithful to \cite{ErGr80}.
The constant $c_0$ is called the Vardi constant.
This is true, and was proved independently by Kamio \cite{Ka25} and Li and Tang \cite{LiTa25}.
References
[ErGr80] Erd\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Ka25] Y. Kamio, Asymptotic analysis of infinite decompositions of a unit fraction into unit fractions. arXiv:2503.02317 (2025).
[LiTa25] Z. Li and Q. Tang, On a conjecture of Erd\H{o}s and Graham about the Sylvester's sequence. arXiv:2503.12277 (2025).
