Ulam numbers: twin pairs, eventual gap-periodicity, and zero density (Erdős #342)
Statement
Define a sequence by $a_1=1$, $a_2=2$, and for $n\geq2$ let $a_{n+1}$ be the least integer greater than $a_n$ that can be written in exactly one way as $a_i+a_j$ with $i<j\leq n$. These are the Ulam numbers, beginning $1,2,3,4,6,8,11,13,16,18,26,28,\ldots$. What can be said about this sequence? In particular: do infinitely many pairs $a,a+2$ (twin Ulam numbers) occur? Is the difference sequence $a_{n+1}-a_n$ eventually periodic? Does the sequence have density $0$?
Acceptance. FULLY RESOLVES: a complete proof settling at least one of the three listed questions — that there are (or are not) infinitely many twin Ulam pairs $a,a+2$; that the gap sequence $a_{n+1}-a_n$ is (or is not) eventually periodic; or that the density of the Ulam numbers is (or is not) $0$. ADVANCES (each independently checkable): (a) a rigorous nontrivial partial result, e.g. a proved positive lower bound on the density, or infinitely many twins under a clearly stated hypothesis; or (b) computation of the Ulam numbers to a new record $N$ with reproducible code, reporting the count, empirical density, twin count, and observed gap/periodicity statistics, improving on the published extent. Deliver a proof, or code plus the attained $N$ and its certified statistics.
Background
A problem of Ulam, recorded by Erdős–Graham [ErGr80, p.53], appearing as problem C4 in Guy's Unsolved Problems in Number Theory [Gu04] and as Problem 7 on Ben Green's open-problems list; listed as open on erdosproblems.com/342 (fetched 2026-07-13, status 'open', tagged 'number theory'). The Ulam numbers form OEIS A002858. Extensive computation indicates the sequence has positive density (empirically about $0.074$), so the answer to 'is the density $0$?' is expected to be no, and analyses of Steinerberger type reveal a surprising hidden near-periodic signal in the distribution of Ulam numbers; nevertheless all three listed questions — infinitude of twins, eventual periodicity of gaps, and whether the density is $0$ — remain unproven. This is the additive cousin of the greedy sequences in Erdős #340 (Mian–Chowla, erdosproblems.com/340), #341, and #359. Erdős attached no prize. The attacker's tool: sieve the Ulam numbers to very large $N$ (maintaining representation counts), measure the empirical density, the frequency of twins, and the gap structure, probe the conjectured hidden periodicity, and convert any structural regularity into a rigorous bound.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #342 (T. F. Bloom) | website |
| REF-02 | OEIS A002858 — Ulam numbers | website |
| REF-03 | B. J. Green, Open problems (Problem 7) | paper |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.