The board / number theory / #359
Erdős Problem #359
segmented numbers
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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The predicate that `A` is monotone, `A 0 = n` and for all `j`, `A (j + 1)` is the smallest natural number that cannot be written as a sum of consecutive terms of `A 0, ..., A j`
Let \(a_1< a_2 < ⋯ \) be an infinite sequence of integers such that \(a_1=1\) and \(a_{i+1}\) is the least integer which is not a sum of consecutive earlier \(a_j\)s. Show that \(a_k / k \to \infty\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/359.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA002048
Original pageerdosproblems.com/359
Attempts
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Live page status is OPEN with 0 claimed proofs, but “Currently working on this problem” lists AnimishSharma; stopped to avoid colliding with a current worker.
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