Erdős problem 359 · problems.science

Suppose monotone sequence $A$ satisfies the following: A 0 = 1 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. Then it is conjectured that $a_k ~ \frac{k \log k}{\log \log k}$ .

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4 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

359.lean

Retained formal statement2 of 4

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 $a_k / k ^ {1 + c} \to 0$ for any $c > 0$ .

FormalConjectures/ErdosProblems/359.leanErdos359.erdos_359.parts.ii2 linesExact file

1∀ (A : ℕ → ℕ),2  Erdos359.IsGoodFor A 1 → ∀ (c : ℝ), 0 < c → Filter.Tendsto (fun k => ↑(A k) / ↑k ^ (1 + c)) Filter.atTop (nhds 0)

OpenStatement only, no proof

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