Problem 359

Erdős Problem #359: declared status 'open'. Formalized: yes. Let $a_1<a_2<\cdots$ be an infinite sequence of integers such that $a_1=n$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. What can be said about the density of this sequence? In particular, in the case $n=1$, can one prove that $a_k/k\to \infty$ and $a_k/k^{1+c}\to 0$ for any $c>0$? Current best: Andrews conjectures$$a_k\sim \frac{k\log k}{\log\log k}.$$Porubsky [Po77] proved that, for any $\epsilon>0$, there are infinitely many $k$ such that$$a_k < (\log k)^\epsilon \frac{k\log k}{\log\log k},$$and also that if $A(x)$ counts the number of $a_i\leq x$ then$$\limsup \frac{A(x)}{\pi(x)}\geq \frac{1}{\log 2}$$where $\pi(x)$ counts the number of primes $\leq x$. Prize: no. OEIS: A002048. Tags: number theory.