Problem 291

../

claims/: The 4 claim pages of Problem 291, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $n\geq 1$ and define $L_n$ to be the least common multiple of $\{1,\ldots,n\}$ and $a_n$ by

$$\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}.$$

Is it true that $(a_n,L_n)=1$ and $(a_n,L_n)>1$ both occur for infinitely many $n$?

Formulation. The site's wording, accessed 2026-09-18 (page last edited 12 January 2026). The fraction $a_n/L_n$ need not be in lowest terms: with $H_n=\sum_{k\le n}1/k=c_n/d_n$ reduced, $d_n$ divides $L_n$ and $(a_n,L_n)=L_n/d_n$, so the first question asks whether the reduced denominator of $H_n$ equals $\mathrm{lcm}(1,\ldots,n)$ for infinitely many $n$ and the second whether it falls short infinitely often. The two are independent questions; the second is answered below, and the status attaches to the pair through the first. The site's example, the integers whose leading digit in base $3$ is $2$, needs $n\ge6$: at $n=2$ the prime $3$ exceeds $n$ and $(a_2,L_2)=(3,2)=1$.

Status. Open: the site's label is OPEN (; page last edited 12 January 2026), and the site marks the problem as not resolvable by a finite computation. The standing derived from the claim pages is open, claim none. Three of the four claim pages concern the second question: Steinerberger's base-3 observation, a pending partial claim answering the second question in the affirmative, which the site's commentary credits to him while the site labels the whole problem OPEN and declares no parts, so that the credit is context and not acceptance, Shiu's criterion and infinitude theorem, a pending partial claim giving the same answer with the exact criterion (the commentary credits the observation, not this paper), and van Doorn's generalization to periodic numerators, a pending partial claim from a dated note with a conditional Lean proof by the prover Aristotle. The fourth, Wu and Yan's conditional density theorem, an accepted conditional claim, gives the second question only under an open independence conjecture and derives nothing; no claim settles or pends on the first. For the first, no proof, disproof, preprint or proof claim that $(a_n,L_n)=1$ infinitely often was found in the search whose scope the Current assessment records: Shiu's paper states it as a conjecture, and Wu and Yan's theorem is conditional and concerns the other half. This is a bounded negative finding, not a certificate of openness.

Source. erdosproblems.com/291, accessed 2026-09-18: the problem page (OPEN, with the site's note that no finite computation can resolve the problem; source keys [ErGr80, p. 34], [Sh16], [WuYa22]; last edited 12 January 2026; OEIS A110566 linked; the statement recorded as formalized), its three-comment discussion thread and its empty proof-claim tab. The site thanks Stefan Steinerberger and Wouter van Doorn. Cite as: T. F. Bloom, Erdős Problem #291, https://www.erdosproblems.com/291, accessed 2026-09-18.

References.

• [ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), printed p. 34. Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [Sh16] Shiu, P., The denominators of harmonic numbers (Revised). arXiv:1607.02863v2 (30 July 2024; v1 of 11 July 2016), 8 pages; an unrefereed preprint. Library home: shiu_2016_denominators_harmonic_numbers_revised; result pages for Theorem 1, Theorem 2 and the Conjecture.
• [WuYa22] Wu, B.-L. and Yan, X.-H., On the denominators of harmonic numbers. IV. C. R. Math. Acad. Sci. Paris 360 (2022), 53--57, DOI 10.5802/crmath.282 (received 11 August 2021, accepted 12 October 2021, published online 26 January 2022). Library home: wu_2022_denominators_harmonic_numbers_iv; result page for Theorem 2.
• [vD24] van Doorn, W., On the non-monotonicity of the denominator of generalized harmonic sums. arXiv:2411.03073 (v2, 23 July 2025); cited in the discussion thread as the source of the generalization below. Library home: doorn_2024_non_monotonicity_denominator_generalized_harmonic_sums.
• [OEIS] Vrabec, F., Sequence A110566, The On-Line Encyclopedia of Integer Sequences (2005; entry revision 39 of 17 May 2023): $a(n)=L_n/d_n=(a_n,L_n)$, with a table to $n=10000$; accessed.

Formalization. Statement only. The file ErdosProblems/291.lean of formal-conjectures at the linked commit (main,) defines L n as the lcm of $1,\ldots,n$ and a n as $\sum_{k\le n}L_n/k$ (with test lemmas for $n\le4$ proved by decide) and declares erdos_291.parts.i : answer(sorry) ↔ {n | gcd (a n) (L n) = 1}.Infinite under category research open and erdos_291.parts.ii : answer(True) ↔ {n | gcd (a n) (L n) > 1}.Infinite under category research solved, both with proof sorry, together with sorry-bodied variants for the leading-digit criterion, for Shiu's heuristic (order $x/\log x$; density zero) and for the Wu--Yan theorem with the independence hypothesis as an explicit argument. The community database (teorth/erdosproblems, data/problems.yaml,) records status open (31 August 2025), a formalized statement (24 June 2026), formal status unformalized and OEIS A110566. Van Doorn's external Lean file, which proves a generalization of the second question under two hypotheses it does not prove, is described under Formalization and external Lean artifact below and has its own claim page; no build of it is recorded.

Current assessment

The question (site formulation, accessed 2026-09-18). The statement above; OPEN, last edited 12 January 2026; source [ErGr80, p. 34]. The commentary credits Steinerberger with the observation that the second question has an easy affirmative answer, the base-$3$ instance being that $3$ divides $(a_n,L_n)$ whenever the leading digit of $n$ in base $3$ is $2$; it states the general criterion, that a prime $p\le n$ divides $(a_n,L_n)$ exactly when $p$ divides the numerator of $H_k$ for $k$ the leading digit of $n$ in base $p$ (the digit $p-1$ always qualifying, by Wolstenholme's theorem), with the one-line reason that $a_n$ is congruent to that numerator's sum modulo $p$; it draws from this, citing Shiu, the heuristic that about $x/\log x$ integers $n\le x$ have $(a_n,L_n)=1$, so that such $n$ should be infinite in number but of density zero, and notes that the heuristic resists proof; and it reports Wu and Yan's theorem that, if the numbers $1/\log p$ are linearly independent over $\mathbb Q$ for every finite set of primes (a consequence of Schanuel's conjecture), the set with $(a_n,L_n)>1$ has upper density $1$. The thread has three comments, described below. The proof-claim tab is empty. The community database record is summarized under Formalization.

Origin. Printed p. 34 of the 1980 monograph, after the $b(a)$ question of Problem 290: "If we set $\sum_{k=1}^n1/k=\frac{a}{L_n}$ where $L_n=\mathrm{lcm}\{1,\ldots,n\}$ then is it true that infinitely often we have $(a,L_n)=1$ and infinitely often we have $(a,L_n)>1$?" The site's statement is this question with $a_n$ for $a$.

The second question: yes. Write $D_n=\mathrm{lcm}(1,\ldots,n)=d_nq_n$ with $H_n=c_n/d_n$ reduced, so that $q_n=(a_n,L_n)$. Shiu's Theorem 2 (arXiv v2, p. 2; claims checked and the half-page proof read through) is the exact criterion: for an odd prime $p\le n$, $p\mid q_n$ if and only if the leading digit $m$ of $n$ in base $p$ satisfies $p\mid c_m$, that is, $mp^a\le n<(m+1)p^a$ for some $a\ge1$ and some $m$ with $p$ dividing the numerator of $H_m$. Since $p\mid c_{p-1}$ for every odd prime (pair $1/j$ with $1/(p-j)$), the leading digit $p-1$ always qualifies; with $p=3$ this is the site's observation, $3\mid(a_n,L_n)$ for $2\cdot3^a\le n<3^{a+1}$, $a\ge1$, a set of positive lower density. So $(a_n,L_n)>1$ holds for infinitely many $n$, and the formal-conjectures file marks this part solved. Shiu's [../library/unit_fractions/shiu_2016_denominators_harmonic_numbers_revised/theorem_1|Theorem 1], $d_n<d_{n-1}$ for infinitely many $n$, gives a second route, since $d_n<d_{n-1}\le D_{n-1}\le D_n$ forces $q_n>1$. Checked by exact arithmetic: the criterion holds for all odd primes $p<60$ and all $n\le3000$ with $p\le n$ (47,578 pairs); $q_n$ is odd for all $n\le10000$ (Shiu's remark that the $2$-adic parts of $d_n$ and $D_n$ agree, also the OEIS entry's comment); and the values $q_n$ for $n\le10000$ agree with the table of OEIS A110566.

The first question: open. No source proves that $d_n=D_n$ (equivalently $q_n=1$) holds infinitely often. By Theorem 2 this says that $n$ avoids, for every odd prime $p\le n$, all the intervals $[mp^a,(m+1)p^a)$ with $p\mid c_m$, a simultaneous leading-digit condition over all primes up to $n$; Shiu writes (p. 6) that "we have yet to discover why there are arbitrarily large $n$ with $d_n=D_n$" and that he could not "emulate Euclid's elegant proof that there are infinitely many primes". His Conjecture (p. 2) is quantitative: $K_1x/\log x<\tilde Q(x)<K_2x/\log x$ for the count $\tilde Q(x)$ of $n\le x$ with $d_n=D_n$, based, as he says (p. 6), on display (3) in the proof of his Theorem 3: $Q_p=\{n:p\mid q_n\}$ has $Q_p(x)\sim|E_p|x/p$ along $x=p^b$, the sieve picture in which each prime removes a proportion about $|E_p|/p$ of the integers, $E_p=\{1<m<p:p\mid c_m\}$. Theorem 3 itself gives $Q_p$ the harmonic density $(\log p)^{-1}\sum_{m\in E_p}\log(1+1/m)$. The site's heuristic is the same in the form of van Doorn's comment (30 November 2025): treating the condition that $p$ does not divide $c_k$, for $k$ the leading digit of $n$ in base $p$, as an event of probability about $1-1/p$ for each $p\le n$ gives, by Mertens' theorem, a proportion about $c/\log n$ of good $n$. Data: Shiu (Section 8) lists 2641 values of $n\le10000$ with $d_n=D_n$, in 26 runs of consecutive integers, recomputed by exact rational arithmetic with the same count and the same 26 runs, except that the last run, printed as $9156_{155}$ (that is, $9156\le n<9311$), has 156 members ($9156\le n\le9311$), so the printed run lengths sum to 2640. The obstacle van Doorn names is the rational independence of the numbers $\log p_1/\log p_i$ needed for Kronecker-type alignment arguments (his MathOverflow question 76372 of 2011 on this independence had no answer on 2026-09-18); he also quotes the bound $|E_p|\le\frac{3^{2/3}}{2}p^{2/3}\approx1.04p^{2/3}$ for the number of $k<p$ with $p\mid c_k$, attributed to himself and, independently, to Lemma 2.4 of Wu and Chen (J. Number Theory 175 (2017); not held in the library, so the bound is second-hand).

The conditional density theorem. Wu and Yan's Theorem 2, the accepted conditional claim Wu and Yan 2022 (C. R. Math. 360 (2022), p. 54, refereed; claims checked and the two-page proof read through): assuming their Conjecture 1, that $1/\log q_1,\ldots,1/\log q_l$ are linearly independent over $\mathbb Q$ for any distinct primes $q_i$ (a consequence of the weak Schanuel conjecture), the set $\mathcal L=\{n:d_n<D_n\}=\{n:(a_n,L_n)>1\}$ has upper asymptotic density $1$. The proof aligns intervals $((p_i-1)p_i^{s_i-1},p_i^{s_i})$, on which $p_i\mid q_n$, with the intervals $(a_ip_2^{\,q},a_{i-1}p_2^{\,q})$ through Kronecker's theorem and Mertens' theorem. It concerns the trivial half and says nothing about infinitude of the complement, but it makes the heuristic that the good set has density zero a consequence of the independence conjecture along a subsequence. Van Doorn's comment writes that under Schanuel the upper density of the set of $n$ with coprime $a_n$ and $L_n$ is $1$; the paper and the site's commentary concern the set with $\gcd>1$, so the comment's sentence appears to name the wrong half. Bloom's comment of 28 December 2025 suggests that any linear dependencies among the $1/\log p$ could be built into the argument, as in his recent work with Croot (arXiv:2509.02835, cited there), possibly giving an unconditional proof of the upper density statement; it makes no proof claim.

Formalization and external Lean artifact. The formal-conjectures file at the pinned commit is summarized under Formalization: statements with sorry bodies, part (ii) marked solved. Van Doorn's comment of 6 February 2026 links ErdosProblem291.lean in the repository Woett/Lean-files (linked at its last change, of 2 March 2026; at that commit 156,824 bytes, importing Mathlib, with no sorry). The file sets, for an integer sequence $(r_i)$, $X_n=L_n\sum_{i\le n}r_i/i$ and proves generalErdos291: if $r$ is periodic with period $t$ and never zero, then under two explicit hypotheses, $m^{2z(m)}<e^{2.52m}$ for all $m\ge4$ with $z(m)$ the number of primes below $m$, and $L_n>2^n$ for all $n\ge100$, for every $N$ there is $b$ with $\gcd(|X_b|,L_b)>N$; the intermediate ohyeah1 gives an $n$ for which $X_n$ has a prime factor at least $m$. The two hypotheses are known theorems that the file takes as hypotheses rather than proving: the first is the Rosser--Schoenfeld bound $\pi(m)\log m<1.26m$ in another form, and the second follows from Nair's lower bound for $\mathrm{lcm}(1,\ldots,n)$. The header says they are prime-number-theorem-type results expected from a separate formalization project, that the proof was produced by Aristotle, Harmonic's automated proving system, from the author's note "Generalized harmonic sums have arbitrarily large prime factors" (uploaded to his repository on 5 February 2026 and linked in the comment, itself based on [vD24]), and that the case $r\equiv1$ is the second question of this problem. The comment adds that for arbitrary (non-periodic) signs $r_i\in\{-1,1\}$ the generalization fails: the signs can be chosen so that $\gcd(X_n,L_n)=1$ for all $n$. The theorem generalizes the settled half and is not progress on the open one; the note and the file are the pending partial claim van Doorn 2026, and no build of the file is recorded.

Claims. Four claim pages: three pending partial claims for the second question, and one accepted conditional claim that gives it only under an open independence conjecture and derives nothing. The partial claims are Steinerberger's base-3 observation, which the curator's commentary credits to him, dated by the earliest archived copy of the site's page that carries it (7 November 2024; the copy of 19 June 2024 has the statement and no commentary), and carrying as a formalization link the Lean package submitted on 16 September 2026 to a prize program's repository, which proves part (ii) from the observation and claims no novelty (no build of it is recorded); the site labels the problem OPEN and declares no parts, so the curator's credit is context and not acceptance, and the claim is pending; Shiu's criterion and infinitude theorem, whose statement the site's commentary records without crediting the paper; and van Doorn's generalization to periodic numerators, from his note of 5 February 2026 with the Lean proof by Aristotle described above, which settles the second question for every periodic sequence of non-zero numerators and so, with $r\equiv1$, the second question itself. The conditional claim is Wu and Yan's conditional density theorem, upper density $1$ for the set with $(a_n,L_n)>1$ under the linear independence over $\mathbb Q$ of the numbers $1/\log q$ for distinct primes $q$. The first question has no claim page, because no source claims a proof or disproof of it. Items recorded on this page without a page of their own, and why: the AI-generated report below claims no result beyond the settled half; Bloom's comment of 28 December 2025 is a suggestion without a manuscript.

Data and AI-assisted items (leads with provenance, not status).

• OEIS A110566 (F. Vrabec, 2005; revision 39, 17 May 2023; accessed): $a(n)=\mathrm{lcm}(1,\ldots,n)/\mathrm{den}(H_n)$, with a table to $n=10000$, the comment that $a(n)$ is always odd, and a conjecture (J. Song, 2022) that every odd number occurs; the table agrees with an exact recomputation.
• A public AI-generated working report on this problem (erdosproblemaday.com/report/291, "access/check date 2026-07-26", labeled PARTIAL) derives the leading-digit criterion, reports an exact sieve computation of $G(N)=\#\{n\le N:(a_n,L_n)=1\}$ with $G(10^k)=7,37,145,2641,20128,138902,615233,10323214$ for $k=1,\ldots,8$ and 409 maximal good intervals in $[1,10^8]$, and states that infinitude is unproved. The hosting site declares the report AI-generated with a named human collaborator and says that it is not an independently verified claim. The values through $10^4$ agree with an exact recomputation; the rest is unverified, and the report proves nothing about the open question.

Search scope. The problem, discussion and proof-claim pages; the community database record; the formal-conjectures file at the pinned commit; the external Lean file above (GitHub API for its last commit; the raw file); the arXiv listing for 1607.02863 (v1 and v2, no journal reference); the Crossref record of DOI 10.5802/crmath.282 and bibliographic title queries (no journal version of Shiu's paper); the Semantic Scholar citation list of the Wu--Yan paper (two records: Wu and Chen, On the denominators of harmonic numbers, III, Period. Math. Hungar. 2023, and a 2024 Mathematika paper on a $p$-divisibility conjecture of Graham; the query for Shiu's paper was rate-limited); arXiv API searches for abstracts naming harmonic numbers and denominators (eleven records, none new on this question) and for "Erdos problem" with the problem number (none); OEIS A110566 and its table; the Stack Exchange API record of MathOverflow question 76372 (unanswered); the AI report above; the GitHub API record of the pull request linked on the Steinerberger claim page; the monograph's p. 34; one general web search (which surfaced the pull request and the report, nothing else). Not searched: MathSciNet, zbMATH, Google Scholar full text, X. Not consulted: Wu and Chen 2017, 2018, 2019 and 2023, Sanna 2016, Boyd 1994, Eswarathasan--Levine 1991 (all cited in the sources), Bloom--Croot arXiv:2509.02835. Nothing found proves or refutes the first question; this is a bounded negative finding.

Remaining gaps. (1) The first question has no proof; Shiu's conjecture and the Mertens heuristic are the only account of it. (2) Wu and Yan's theorem is conditional on an open independence conjecture; the suggested unconditional route is a comment. (3) Shiu's paper is an unrefereed preprint; its Theorems 3 and 4 are compiled as claims only. (4) The external Lean file proves its theorem under two hypotheses it does not prove, and no build of it is recorded. (5) The $1.04p^{2/3}$ bound for $|E_p|$ is second-hand.

Progress and known results

• Erdős and Graham (1980, printed p. 34): the question.
• Second question, yes: the leading-digit criterion, Shiu's Theorem 2 (preprint, 2016/2024; the pending partial claim Shiu 2016), with Steinerberger's observation (the pending partial claim Steinerberger 2024) as its case $m=p-1$, $p=3$; also Shiu's [../library/unit_fractions/shiu_2016_denominators_harmonic_numbers_revised/theorem_1|Theorem 1]. The denominator drops $d_n<d_{n-1}$ are the $a=1$ case of Problem 290. Van Doorn's generalization to periodic non-zero numerators, with a Lean proof by Aristotle under two classical hypotheses (the pending partial claim van Doorn 2026).
• Conditional: Wu and Yan's Theorem 2 (2022, refereed; the accepted conditional claim Wu and Yan 2022): under Conjecture 1, the set with $(a_n,L_n)>1$ has upper density $1$.
• First question, open: Shiu's Conjecture, $\tilde Q(x)\asymp x/\log x$; 2641 good $n\le10000$ (Shiu, recomputed); OEIS A110566.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1980_old_new_problems_results_combinatorial_number_theory
• doorn_2024_non_monotonicity_denominator_generalized_harmonic_sums
• shiu_2016_denominators_harmonic_numbers_revised
• shiu_2016_denominators_harmonic_numbers_revised / conjecture_p2
• shiu_2016_denominators_harmonic_numbers_revised / theorem_1
• shiu_2016_denominators_harmonic_numbers_revised / theorem_2
• shiu_2016_denominators_harmonic_numbers_revised / theorem_3
• shiu_2016_denominators_harmonic_numbers_revised / theorem_4
• wu_2022_denominators_harmonic_numbers_iv
• wu_2022_denominators_harmonic_numbers_iv / theorem_2