PROVED
This has been solved in the affirmative.
Let $n\in\mathbb{N}$ with $n\neq p^k$ for any prime $p$ and $k\geq 0$. What is the largest integer not of the form\[\sum_{1\leq i<n}c_i\binom{n}{i}\]where the $c_i\geq 0$ are integers?
If $n=\prod p_k^{a_k}$ then the largest integer not of this form is\[\sum_k \left( \sum_{1\leq d\leq a_k}\binom{n}{p_k^d}\right)(p_k-1)-n.\]This was first proved by Hwang and Song
[HwSo24]. Independently this was found in the comment section by Peake and Cambie.
The sequence of such thresholds is
A389479 in the OEIS.
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This page was last edited 13 October 2025.
Additional thanks to: Boris Alexeev, Stijn Cambie, and Michael Peake
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #435, https://www.erdosproblems.com/435, accessed 2026-01-16
