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Claim. Let $a_n$ be the sequence of Problem 423 and $b_n=a_n-n$. Quanyu Tang, The Hofstadter consecutive-sum sequence omits infinitely many positive integers, arXiv:2603.09939 (carded at its library home), proves in its version 2 of 23 March 2026:

• Theorem 1.3: $(b_n)$ is nondecreasing and unbounded, so $a_n=n+\omega(1)$ and the sequence omits infinitely many positive integers, the conjecture recorded in the comments of OEIS A005243;
• Theorem 1.4: $a_n\ge n+\log\log n/\log20-O(1)$;
• Theorem 1.5: for every $\varepsilon>0$ there is $C_\varepsilon$ with $a_n\le C_\varepsilon n^{4175/2506+\varepsilon}$ for all $n\ge1$.

Monotonicity is immediate from $a_{n+1}\ge a_n+1$; unboundedness uses that a positive integer is a sum of at least two consecutive positive integers exactly when it is not a power of $2$, together with the finiteness, from the Schinzel-Tijdeman theorem, of the solutions of $v^2+v+E=2^k$ for fixed $E$. The upper bound writes the terms as differences of prefix sums, whose set is convex, and applies Bloom's lower bound $|A-A|\gg_\eta|A|^{6681/4175-\eta}$ for finite convex sets $A$; the exponent is $1/(c-1)$ for $c=6681/4175$, and the same argument gives $a_n\ll n^{1/(c-1)+o(1)}$ from any bound $|A-A|\ge|A|^{c-o(1)}$ for convex sets, the form in which the site's commentary states the result.

The result grew in four postings, all linked above. The note of 15 January 2026 in Tang's repository, announced in the site's thread the same day, proved the first statement (its Theorem 1.3). Version 1 of the arXiv paper, of 10 March 2026, added the upper bound (its Theorem 1.6) and extended the first statement to every finite seed (Theorem 1.4 and Corollary 1.5); its lower bound is only $n+\omega(1)$. The $\log\log n$ lower bound, which Yann Bugeaud suggested could be extracted from the method, came in the repository's note of 13 March 2026 (its Theorem 1.6) and in version 2, whose Theorems 1.3-1.5 are the three statements above. Section 6.2 of version 2 records Sothanaphan's observation that Cushman's bound $|A-A|\gg_\varepsilon|A|^{8/5+1/3440-\varepsilon}$ for convex sets gives $a_n\ll_\varepsilon n^{688/413+\varepsilon}$ by the same argument, the bound the site's commentary records; it is an observation recorded in the paper, not one of its theorems. The conjecture of Erdős and Hegyvári that $c=2$ is admissible in the convex-set problem would give $a_n\le n^{1+o(1)}$; that is a hypothesis, not a result. Tang's thread post of 11 March 2026 says the work made exploratory use of ChatGPT-5.2 Thinking, with every argument checked by the author, and that the system first produced a polynomial bound with a large exponent, which the author then improved; the paper's declaration limits the AI assistance to brainstorming for Theorem 1.5 and drafting the code for its figure. Tang is the claimant, with the system named as Tang names it. Bolan's independent proof of the first statement, acknowledged in the paper, is on its own page.

Covers. $a_n-n$ is nondecreasing and unbounded, so the sequence omits infinitely many positive integers; $a_n\ge n+\log\log n/\log20-O(1)$; and $a_n\le C_\varepsilon n^{4175/2506+\varepsilon}$ for every $\varepsilon>0$. Not covered: the asymptotic behavior the problem asks for, which the paper's Problem 6.1 ($a_n=n+o(n)$?) leaves open.

Depends on. No page of this wiki.

Standing. Claimed: the arXiv record calls version 2 the submitted version, and no refereed version exists; the site's commentary (page last edited 23 March 2026) credits the three results to Tang, but commentary on a problem the site labels OPEN is not acceptance. The claim stays claimed.