New Bounds for Sequences with Distinct Consecutive Sums in Erdős Problem 357

Jeff Pickhardt

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New Bounds for Sequences with Distinct Consecutive Sums in Erdős Problem 357

Abstract

In Erdős Problem 357, let 𝑓(𝑛) be the largest integer 𝑘 for which there are integers 1 ≤ 𝑎_1 < ⋯ < 𝑎_𝑘 ≤ 𝑛 such that all sums of consecutive terms 𝑎_𝑢 + ⋯ + 𝑎_𝑣 are distinct. Erdős and Harzheim asked for the order of 𝑓(𝑛) and, in particular, whether 𝑓(𝑛) = 𝑜(𝑛). The previously recorded bounds are (2 + 𝑜(1))√𝑛 ≤ 𝑓(𝑛) ≤ ( 2 1 3 512 + 𝑜(1)) 𝑛. We improve both constants. A construction obtained by deleting one residue class modulo 3 gives 𝑓(𝑛) ≥ ( 4 √3 − 𝑜(1)) √𝑛, while a layer-packing argument that uses the increasing order gives 𝑓(𝑛) ≤ 𝑛 2 + 𝑂(𝑛2/3) = ( 1 2 + 𝑜(1)) 𝑛.

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