Erdős problem 434

Let $k \le n$. What choice of $A\subseteq\{1, \dots, n\}$ (with $\text{gcd}(A) = 1$) of size $|A| = k$ maximises the number of integers not representable as the sum of finitely many elements from $A$ (with repetitions allowed)? Is it $\{n, n - 1, \dots, n - k + 1\}$?

The maximal choice is indeed $\{n, \dots, n - k + 1\}$, as proved by Kiss [Ki02].

The Lean theorem assumes $2 \le k$. When $k = 1 < n$, the proposed singleton $\{n\}$ does not have gcd $1$; the gcd condition instead forces the unique admissible choice $A = \{1\}$.