Problem Statement
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 $c>0$?
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 $c>0$?
Categories:
Number Theory
Progress
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\[a_k\sim \frac{k\log k}{\log\log k}.\]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 $\leq x$.See also [839].
Source: erdosproblems.com/359 | Last verified: January 14, 2026