---

Claim. For integers $2\le k\le n$, among the sets $A\subseteq\{1,\ldots,n\}$ with $|A|=k$ and $\gcd(A)=1$, the set $\{n-k+1,\ldots,n\}$ maximizes the number of positive integers that are not sums of elements of $A$ with repetition. This is Theorem 1 of G. Kiss, On the extremal Frobenius problem in a new aspect, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 45 (2002), 139--142, in the paper's notation $\nu(n,t)=N(t-n+1,\ldots,t)$ for $1<n\le t$; its Lemma counts the gaps of that set as $(t-n+r-1)q/2$ where $t=q(n-1)+r$ with $1\le r\le n-1$. The paper prints "Received February 13, 2003", the page's date, and no later publication date; its running head prints volume 44 where the archive files it as Tomus XLV. This answers the second question of Problem 434, whether $\{n,n-1,\ldots,n-k+1\}$ is the maximizing choice, with yes, it is a maximizer. It is in general not the only one: Theorem 2 of the same paper (§3, Optimal sets, p. 141) exhibits a second optimal set in many cases. In the problem's notation, write $n=dk+r$ with integers $2\le d<k$ and $0\le r<k-d$; if $k-r\equiv0$ or $-1\pmod{d+1}$, then at least two admissible $k$-element sets attain the maximum, the second consisting of the multiples of $d+1$ up to $n$ together with the largest elements of one residue class modulo $d+1$ (the class of $-1$ in the first case, of $1$ in the second), whose gap count Sylvester's formula gives as the Lemma's value. Boris Alexeev noted in the site's thread on 2025-10-29 that the maximizer is often not unique, and the site marks the comment as addressed. So the first question, read as asking for a maximizing set, is answered by the top block; Theorem 2 shows that other sets also maximize in its cases, other maximizers occur outside them as well (for $n=5$ and $k=3$, $\{2,4,5\}$ leaves $1$ and $3$ unrepresented, as many integers as $\{3,4,5\}$ leaves), and the full set of maximizers is not determined. The problem's wording allows $k=1$, where the only admissible set is $\{1\}$, with no gaps, and the proposed set $\{n\}$ is admissible only for $n=1$; Kiss's hypothesis and the formal-conjectures statement both read the question with $k\ge2$, as this page does. The site's commentary cites the paper from a thread post of 2025-10-29 and thanks the poster. This corpus has not checked the paper's proofs step by step.

Depends on. Dixmier's theorems: Kiss's proof applies Theorem 2 of Dixmier's paper, that for $j\ge1$ the interval $((j-1)n,jn]$ contains at least $\min(n,\,j(k-1)+1)$ integers representable by $A$, to compare the gaps of an arbitrary admissible set with those of $\{n-k+1,\ldots,n\}$ interval by interval.

Acceptance. The paper is a journal publication in the Annales Universitatis Scientiarum Budapestinensis, Sectio Mathematica, the refereed evidence. The site's curator, T. F. Bloom, labels the problem proved and credits Kiss's paper with the proof; that documented acceptance is the reviewed evidence. The Lean suffix of the site's label refers to a separate AI-assisted Lean proof that argues from Dixmier's theorem without Kiss's count; it is recorded as the claimed page Lean proof through Dixmier's interval theorem and supplies no evidence here.