Problem 357

Erdős Problem #357: declared status 'open'. Formalized: yes. Let $1\leq a_1<\cdots <a_k\leq n$ be integers such that all sums of the shape $\sum_{u\leq i\leq v}a_i$ are distinct. Let $f(n)$ be the maximal such $k$. How does $f(n)$ grow? Is $f(n)=o(n)$? Current best: If $g(n)$ is the maximal $k$ such that there are $1\leq a_1,\ldots,a_k\leq n$ with all consecutive sums distinct (i.e. we drop the monotonicity assumption in the definition of $f$) then Hegyv\'{a}ri [He86] has proved that$$\left(\frac{1}{3}+o(1)\right) n\leq g(n)\leq \left(\frac{2}{3}+o(1)\right)n.$$The upper bound of Coppersmith and Phillips in [867] implies$$g(n) \leq \left(\frac{2}{3}-\frac{1}{512}+o(1)\right)n.$$A similar question can be asked if we replace strict monotonicity with weak monotonicity (i.e. Prize: no. OEIS: A364132, A364153. Tags: number theory.