Problem 295

Erdős Problem #295: declared status 'open'. Formalized: yes. Let $N\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\leq n_1<\cdots <n_k$ with$$1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$$Is it true that$$\lim_{N\to \infty} k(N)-(e-1)N=\infty?$$ Prize: no. OEIS: A192881. Tags: number theory, unit fractions.