Problem 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 $0$?
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 $0$?
Categories:
Number Theory
Progress
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].
Source: erdosproblems.com/342 | Last verified: January 14, 2026