With $a_{1} = 1$ and $a_{2} = 2$ let $a_{n + 1}$ for $n \geq 2$ be the least integer $> a_{n}$ which can be expressed uniquely as $a_{i} + a_{j}$ for $i < j \leq n$ .What can be said about this sequence? Do infinitely many pairs $a, a + 2$ occur? Does this sequence eventually have periodic differences? Is the density $0$ ?

unreviewedOpen. Best to date: honest null, not yet machine-sealed.

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

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Evidence1

honest null

unverified claim

attempted via frontier 'sidon/B2' (transfer_strength=none) -> no_progress

No solve/partial on this pass. Transfer into the owned frontier was 'none'. 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

This is the **classical Ulam sequence** [[nomath]](the $(1,2)$-Ulam numbers)[[/nomath]], OEIS **A002858**. It begins [ 1,2,3,4,6,8,11,13,16,18,26,28,\dots ] and is defined exactly as you wrote: each new term is the smallest integer larger than the previous term that has **exactly one** representation as a sum of two **…

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

candidate solution ↗

Formal proof

AMS 5 11 40 · test (literature)

theorem erdos_342.test.a0 : ∀ a : ℕ → ℕ, IsUlamSequence a → a 0 = 1

formal-conjectures/342.lean ↗

OEIS1

A002858 — Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms.1,2,3,4,6,8,11,13,16,18,26,28,36,38,47,48,53,57,62,69,72,77,82,87,97,99,102,106,114,126,131,138,145,148,155,175,177,180,

Connections1

open problems list · paper

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