The board / number theory / #839
Erdős Problem #839
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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A sequence \(a : \mathbb{N} \to \mathbb{N}\) is "sum-of-consecutive-free" if no term equals the sum of a contiguous block of earlier terms. That is, for all \(i\), \(a_i \neq a_j + a_{j+1} + \cdots + a_k\) for any \(j \leq k < i\).
Erdős Problem 839 (Part 1) [Er78f][Er92c]: Let \(1 \leq a_1 < a_2 < \cdots\) be a strictly increasing sequence of positive integers such that no \(a_i\) is the sum of consecutive \(a_j\) for \(j < i\). Is it true that \(\limsup a_n / n = \infty\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/839.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/839
Attempts
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exact standard-library search certifies the finite extremal table \(f(N)\) for every \(1\leq N\leq70\), audits the elementary \(2/3\) bound and Freud construction, and isolates uncontrolled overlap am
PARTIAL