Let $a_{1} < a_{2} < \dots$ be an infinite sequence of integers such that $a_{1} = n$ and $a_{i + 1}$ is the least integer which is not a sum of consecutive earlier $a_{j}$ s. What can be said about the density of this sequence?In particular, in the case $n = 1$ , can one prove that $a_{k} / k \to \infty$ and $a_{k} / k^{1 + c} \to 0$ for any $c > 0$ ?

Open — best to date is a honest null, not yet sealed.

number theory · open · formalized (Lean) · 1 attempt

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

honest null

needs verification

attempted via frontier '?' (transfer_strength=n/a) -> no_progress

No solve/partial on this pass. Transfer into the owned frontier was 'n/a'. Do not re-attack cold; needs a new idea or richer accumulated context.

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

and for each (i\ge 1), let (a_{i+1}) be the **least integer (>a_i)** that is **not** representable as a sum of **consecutive** earlier terms (a_j+\cdots+a_k) with (1\le j\le k\le i). [[nomath]](This “$>a_i$” is implicit in the “infinite increasing sequence” condition; otherwise the rule could pick something $\le a_i$.)…

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_359.parts.i (A : ℕ → ℕ) (hA : IsGoodFor A 1) :
    atTop.Tendsto (fun k ↦ (A k : ℝ) / k) atTop

formal-conjectures/359.lean ↗

oeis

A002048 — Segmented numbers, or prime numbers of measurement.1,2,4,5,8,10,14,15,16,21,22,25,26,28,33,34,35,36,38,40,42,46,48,49,50,53,57,60,62,64,65,70,77,80,81,83,85,86,90,91,92,10

status

open

evidence

formal

oeis

status

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