Problem Statement
Is there some $c>0$ such that, for all sufficiently large $n$, there exist integers $a_1<\cdots<a_k\leq n$ such that there are at least $cn^2$ distinct integers of the form $\sum_{u\leq i\leq v}a_i$?
Categories:
Number Theory
Progress
This fails for $a_i=i$ for example. Erdős and Graham also ask what happens if we drop the monotonicity restriction and just ask that the $a_i$ are distinct. They speculated that perhaps some permutation of $\{1,\ldots,n\}$ has at least $cn^2$ such distinct sums - this is true, as proved by Konieczny [Ko15] (see [34]).The original problem was solved (in the affirmative) by Beker [Be23b].
They also ask how many consecutive integers $>n$ can be represented as such a sum? Is it true that, for any $c>0$ at least $cn$ such integers are possible (for sufficiently large $n$)?
See also [34], [357], and [358].
Source: erdosproblems.com/356 | Last verified: January 14, 2026