---

Claim. Let $g(n)$ be the largest $k$ for which there are integers $1\le a_1,a_2,\ldots,a_k\le n$, not required to be increasing, whose sums $\sum_{u\le i\le v}a_i$ over runs of consecutive terms are all different. Hegyvári's Theorem 1 states that

$$\Bigl(\frac13+o(1)\Bigr)n\le g(n)\le\Bigl(\frac23+o(1)\Bigr)n .$$

The paper is Hegyvári, N., On consecutive sums in sequences, Acta Math. Hungar. 48 (1986), no. 1--2, 193--200; the theorem is on printed p. 193 and is recorded on the result page Theorem 1. The lower bound comes from a sequence whose partial sums form a Sidon set, and the upper bound from counting the sums of at most $t$ consecutive terms for a fixed large $t$. Every increasing sequence counted by the function $f(n)$ of Problem 357 is counted by $g(n)$, so $f(n)\le g(n)\le(2/3+o(1))n$. The paper's introduction records the conjecture of Erdős and Harzheim that a linear number of terms is impossible when the sequence is increasing, which is the problem's question, and leaves it open. The formal-conjectures file for the problem, at its revision of 30 September 2026, marks its variant erdos_357.variants.hegyvari as research solved.

Covers. The upper bound $f(n)\le(2/3+o(1))n$. The lower bound concerns $g(n)$ only, since its sequence is not increasing. The result does not answer whether $f(n)=o(n)$, the problem's question.

Acceptance. The refereed evidence is the journal publication cited above, in Acta Mathematica Hungarica. The site labels the problem OPEN, so its commentary credits the paper without settling the problem and no reviewed evidence is listed. The page is dated by the paper's received date as printed, 2 October 1984.

Depends on. The result page Hegyvári's Theorem 1.