OPEN
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 $c>0$?
A problem of MacMahon, studied by Andrews
[An75]. When $n=1$ this sequence begins
\[1,2,4,5,8,10,14,15,\ldots.\]
This sequence is
A002048 in the OEIS.
Andrews conjectures
\[a_k\sim \frac{k\log k}{\log\log k}.\]
See also [839].
