Density of Ulam Numbers

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $u_1=1$ and $u_2=2$. For each $k\geq3$, let $u_k$ be the least integer strictly greater than $u_{k-1}$ for which exactly one pair of indices $1\leq i<j<k$ satisfies $u_i+u_j=u_k$. Thus order does not distinguish two representations and repetition of the same earlier term is forbidden. Put $U=\{u_k:k\geq1\}$. Its natural density, when it exists, is \[ d(U)=\lim_{N\to\infty}\frac{\#(U\cap\{1,\ldots,N\})}{N}. \]

Statement recorded in the supplied source.

Does the natural-density limit $d(U)$ exist, and is it strictly positive? [S2]

Short English statement

Do the classical Ulam numbers occupy a well-defined positive fraction of the positive integers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1270 (NT-063). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Stefan Steinerberger, A hidden signal in the Ulam sequence, arXiv:1507.00267v6 (2016), abstract and Section 1. https://arxiv.org/pdf/1507.00267.