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Claim. Write $\sum_{a\le n\le b}1/n=u_{a,b}/v_{a,b}$ in lowest terms. For every integer $a\ge1$ there is $b>a$ with $v_{a,b+1}<v_{a,b}$, and the least such $b$ is at most linear in $a$. In van Doorn's convention, where $b(a)$ is the least $b>a$ with $v_{a,b}<v_{a,b-1}$ (one more than the site's $b(a)$), van Doorn's Corollary 1 gives $b(a)\le6(a-1)$ for all $a>1$, from the $3$-adic valuation of the block ending at $2\cdot3^{k+1}$ when $3^k<a\le3^{k+1}$, and Theorem 2 gives $b(a)\le4.374(a-1)$ for all $a\ge6$. In the other direction Theorem 6 gives $\liminf_{a\to\infty}(b(a)-a)/\log a\ge1/2$ for every periodic integer numerator sequence, and Theorem 8 places the limit inferior for the classical case strictly between $0.54$ and $0.61$. So the answer to the existence question is yes, and $b(a)$ grows linearly: $a+0.54\log a<b(a)<4.374a$ for all large $a$. The problem asks a yes-or-no question and a growth question; the paper answers the first with a proof and the second with a two-sided estimate, so the value is solved. The same paper settles the existence question for every periodic numerator sequence that is not identically zero (Corollary 2 and Theorem 5); the case $a=1$ had been settled earlier by Shiu (card).

Sources. The paper's card van Doorn 2024 records the arXiv v2 text (23 July 2025, 57 pages), of which no file is held, with result pages for Corollary 1, Theorem 2, Theorem 6 and Theorem 8. Read depth: claims checked for Corollary 1 and Theorems 2, 6 and 8; the proofs of Theorems 1 and 6 are compiled for structure only and the computer-checked table behind Theorem 2 is not rerun. The problem page records a consistency check of $b(a)$ for $a\le66$ against OEIS A375081.

Acceptance. The site's curator, Thomas Bloom, labels the problem proved and credits its resolution to this paper on the problem page (last edited 28 December 2025); the curator is not an author, and that credit is the reviewed evidence. The paper has no journal version (the arXiv record and a Crossref query,), so refereed is not listed. Three Lean files in the author's repository accompany the paper and are linked above as formalizations of its results: ErdosProblem290.lean, written by van Doorn with the prover Aristotle and finished and cleaned up by Boris Alexeev, proves that for every $a>0$ some $b$ with $a<b\le6a$ has $v_{a,b}<v_{a,b-1}$, without sorry, its closing comment listing the axioms propext, Classical.choice and Quot.sound; ErdosProblem290lower.lean proves, without sorry and with no declared axiom, that for all large $a$ the denominator does not drop at any $b<a+\log a/20$, a weaker constant than Theorem 6's $1/2$; and ErdosProblem290lowertight.lean proves a converse for a periodic numerator sequence of period $t$, infinitely many drops at some $b<a+(1+\varepsilon)t(t+1)\varphi(t)\log a$, under one declared axiom that the author says follows from the prime number theorem in arithmetic progressions. None of these files was built or audited by this corpus, so formalized is not listed; the formal-conjectures file for the problem is a statement with a sorry body and is not a formalization. The finer growth of $b(a)-a$ is the subject of the author's two later partial claims, the exact lower limit and the almost-all lower bound.