Erdős Problem #435

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.