Problem 342

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Statement. 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 $0$?

Status. Open.

Source. erdosproblems.com/342, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #342, https://www.erdosproblems.com/342.

References.

• [Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section C4 "Ulam numbers", p. 166: the U-numbers and five questions on them, introduced as some of those Recamán asked, among them positive density (marked as Ulam's), infinitely many consecutive pairs, and arbitrarily large gaps, with Muller's 20000 terms. Library home: guy_2004_unsolved_problems_number_theory.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

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• guy_2004_unsolved_problems_number_theory