Problem 293

Erdős Problem #293: declared status 'open'. Formalized: no. Let $k\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to$$1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$$with $1\leq n_1<\cdots <n_k$. Estimate the growth of $v(k)$. Current best: An elementary inductive argument shows that $n_k\leq ku_k$ where $u_1=1$ and $u_{i+1}=u_i(u_i+1)$, and hence$$v(k) \leq kc_0^{2^k},$$where$$c_0=\lim_n u_n^{1/2^n}=1.26408\cdots$$is the 'Vardi constant' (small improvements on this are possible as in [148]). Prize: no. Tags: number theory, unit fractions.