OPEN
This is open, and cannot be resolved with a finite computation.
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 $0$?
A problem of Ulam. The sequence is\[1,2,3,4,6,8,11,13,16,18,26,28,\ldots\]at
OEIS A002858.
See also Problem 7 of Green's
open problems list.
This is problem C4 in Guy's collection
[Gu04].
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This page was last edited 30 September 2025.
Additional thanks to: Desmond Weisenberg
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #342, https://www.erdosproblems.com/342, accessed 2026-01-14
