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.