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on Upper density of the Ulam sequence

The statement

The Ulam sequence is the unique sequence of positive integers defined as follows. Let a unique Ulam sum for a sequence $a : \mathbb{N} \to \mathbb{N}$ at stage $n$ be an integer $m$ that can be written uniquely as $a(i) + a(j)$ with $i < j < n$ . The Ulam sequence is the sequence $a$ with $a(0) = 1$ , $a(1) = 2$ , and for each $n \geq 2$ , the value $a(n)$ is the least integer greater than $a(n-1)$ that is a unique Ulam sum at stage $n$ . For a subset $S \subseteq \mathbb{N}$ , the upper density is defined as $\limsup_{N \to \infty} \frac{|S \cap \{1, 2, \ldots, N\}|}{N}$ . Determine whether the upper density of the range of the Ulam sequence equals $0$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.