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Claim. Let $a_n$ be the sequence of Problem 423. Matthew Bolan, Hofstader Ulam sequence [sic], a note dated January 2026 in Bolan's repository mjtb49/HofstaderUlam, posted in the site's thread on 15 January 2026, proves as its Theorem 1 that for every $N>0$ the sequence does not contain every integer of the interval $[N,2^{3N+2}]$; so the sequence omits infinitely many positive integers, equivalently $a_n-n$ is unbounded. The proof supposes the interval filled, so that the $N+1$ powers of $2$ from $2^{2N+2}$ to $2^{3N+2}$ are each a sum of consecutive terms; no power of $2$ is a sum of two or more consecutive integers, so each such sum must use terms below $N$, and by the pigeonhole principle two of the powers use the same least term. An elementary lemma on the equation $2^b c=(n+1)+\cdots+m$ then bounds the lengths of the two sums and yields a contradiction. The thread post says the argument gives a lower bound of only about $n+\log^*n$, with $\log^*$ the iterated logarithm, and, after reading Tang's note, that the proof adds nothing beyond some explicit constants. The site's commentary and the acknowledgments of Tang's paper record the two proofs as independent; Tang's result, with its quantitative bounds, is on Tang's page.

Submission note. Posted to the site's forum by Matthew Bolan on 15 January 2026:

I also recently obtained a proof of this, in perhaps a slightly different way. Here is my own short note: https://github.com/mjtb49/HofstaderUlam/blob/main/HofstaderUlamSequence.pdf . My lower bound is terrible, I get something like $n + \log^*(n)$ where $\log^*$ is the iterated logarithm.

EDIT: Now that I've looked at Quanyu Tang's note, I think there is nothing new in my proof besides maybe some constants I kept explicit.

Covers. The sequence omits infinitely many positive integers, equivalently $a_n-n$ is unbounded: for every $N>0$ some integer of $[N,2^{3N+2}]$ is missing. Not covered: the asymptotic behavior the problem asks for.

Depends on. No page of this wiki.

Standing. Claimed: an unrefereed note in a public repository; the site's commentary (page last edited 23 March 2026) credits Bolan and Tang independently with the result, but commentary on a problem the site labels OPEN is not acceptance. The claim stays claimed.