---

Claim. Let $A(n)=\{a_0<a_1<\cdots\}$ be the greedy sequence of Problem 271, with $a_0=0$, $a_1=n$ and each later term the least integer above its predecessor that creates no three-term arithmetic progression. For $n=3^m$, Theorem 1 of A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with the greedy algorithm, Bell Laboratories internal memorandum, January 1978 (card Odlyzko and Stanley 1978), states that a positive integer $t=\sum_i t_i3^i$ belongs to $A(3^m)$ exactly when $t_i\in\{0,1\}$ for $i\ne m$, $t_m=0$ forces $t_{m-1}=\cdots=t_0=0$, and $t_m=2$ forces $\sum_{i<m}t_i>0$. For $n=2\cdot3^m$, Theorem 2 states that $t\in A(2\cdot3^m)$ exactly when $t_i\in\{0,1\}$ for $i\ne m,m+1$, $t_m\in\{0,2\}$, and $t_{m+1}=2$ forces $t_m=0$ and $\sum_{i<m}t_i>0$. As printed, Theorem 2 is false for every $m\ge1$: its conditions admit $t=1$, which is smaller than $a_1=2\cdot3^m$ and so not in $A(2\cdot3^m)$. The case $n=1$ is $m=0$: the members of $A(1)$ are the integers whose ternary expansion has no digit $2$, so $a_k$ is $k$ written in binary and read in ternary (Remark 1). Remark 2 gives the growth for these values of $n$: with $\alpha=\log_23$,

$$\liminf_{k\to\infty}\frac{a_k}{k^{\alpha}}=\frac12 \qquad\text{and}\qquad \limsup_{k\to\infty}\frac{a_k}{k^{\alpha}}=1.$$

The memorandum says that Theorems 1 and 2 can be proved by a routine but tedious case-by-case analysis (p. 2) and gives no proof. For every other $n$ it conjectures, on a probabilistic heuristic and a table of values, growth of order $k^2/\log k$, which it does not prove.

Covers. The values $n=3^m$ and $n=2\cdot3^m$, $m\ge0$: both questions, the explicit determination of the $a_k$ and their rate of growth, are addressed for these $n$, though for $n=2\cdot3^m$ with $m\ge1$ the printed description is false as stated. Not covered: every other $n$, for which no explicit description is known, and the conjectured growth $k^2/\log k$.

Depends on. Nothing in this wiki.

Standing. Claimed. The memorandum states Theorems 1 and 2 without proof, is unrefereed, and no outside review of it is recorded, so no evidence kind is listed; the site labels the problem OPEN, and its commentary credits the descriptions to the memorandum as commentary on an open problem. The descriptions for $m\ge1$ were later proved, with proof, by Rolnick in a refereed paper, recorded as an accepted partial claim on his claim page. The claim is partial and derives nothing for the problem's standing.