Problem 673

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claims/: The 2 claim pages of Problem 673, one per claimant's result; the problem's standing derives from them.

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Statement. Let $1=d_1<\cdots <d_{\tau(n)}=n$ be the divisors of $n$ and

$$G(n) = \sum_{1\leq i<\tau(n)}\frac{d_i}{d_{i+1}}.$$

Is it true that $G(n)\to \infty$ for almost all $n$? Can one prove an asymptotic formula for $\sum_{n\leq X}G(n)$?

Status. The site labels the problem PROVED; its remarks record Tao's bounds $\tau(n/m)/m\le G(n)\le\tau(n)$, displayed for any divisor $m$ of $n$ and true for $m>1$, answering the first question yes, and Erdős's 1982 remark that the divergence is trivial. The standing in the frontmatter derives from the claim pages under the parts divergence and asymptotic: the accepted full claim on Erdős and Tenenbaum's 1983 paper settles both questions, its Théorème 2 giving $\sum_{n\le x}G(n)=x\log x\,(1+o(1))$, and the accepted partial claim on Erdős's remark settles the first; the problem is solved with the value proved.

Source. erdosproblems.com/673, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #673, https://www.erdosproblems.com/673.

References.

• [Er79e] Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73-82.
• [Er82e] Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59-79.
• P. Erdős and G. Tenenbaum, Sur les diviseurs consécutifs d'un entier. Bull. Soc. Math. France 111 (1983), 125--145, doi:10.24033/bsmf.1981.
• Tenenbaum, Gérald, Some of Erdős' unconventional problems in number theory, thirty-four years later (2013), 651--681.

Formalization. Statement in formal-conjectures, added on 2026-09-20 and amended on 2026-09-22 to require $m>1$ in Tao's bound. As amended on 2026-09-22, the file states three results tagged research solved: erdos_673.parts.i, that for every $C$ the set of $n$ with $C<G(n)$ has density $1$; erdos_673.parts.ii, that $\sum_{1\le n\le X}G(n)\sim X\log X$; and erdos_673.variants.average, that the average of $G$ over $n\le X$ tends to infinity. Each carries a formal_proof attribute pointing to the theorem erdos_673 of Boris Alexeev's repository of formalized Erdős problems at a pinned commit, and Tao's bound is stated as a textbook variant. A statement file is not a proof: its own proofs are placeholders. Alexeev's development, which proves both the divergence for almost all $n$ and the asymptotic formula, is linked at a pinned commit from Erdős and Tenenbaum's claim page as a formalization of their result. This corpus built it at that commit, checked that its theorem Erdos673.erdos_673 uses only the axioms propext, Classical.choice and Quot.sound and matches its comparator challenge, and audited the statement, so the claim page lists it as formalized evidence.

Current assessment

The question has two parts. The first, whether $G(n)\to\infty$ for almost all $n$, is answered yes: for a divisor $m>1$ of $n$ every divisor $d$ of $n/m$ is followed in the ordered divisor list by a divisor at most $dm$, so $G(n)\ge\tau(n/m)/m$, and with $G(n)\le\tau(n)$ this makes $G(n)$ comparable to $\tau(n)$ for every $n$ with a prime factor below a fixed bound; the site records the argument as Tao's observation, and Erdős's 1982 survey (Chapter II, section 6, p. 66) calls the divergence trivial while giving no proof. Erdős and Tenenbaum's 1983 paper gives the same argument on p. 127, with $m$ the least prime factor of $n$. The accepted claims are theirs, which settles both parts, and Erdős's remark. The second part, an asymptotic formula for $\sum_{n\le X}G(n)$, was answered by Erdős and Tenenbaum, Bull. Soc. Math. France 111 (1983), 125--145, whose Théorème 2 gives $\sum_{n\le x}G(n)=x\log x\,(1+o(1))$ with an explicit error term; Tenenbaum's 2013 survey sharpens it to $x\log x-\sum_{n\le x}G(n)\asymp x(\log x)^{1-\delta}/(\log\log x)^{3/2}$, where $\delta=1-\log(e\log2)/\log2$. Erdős's 1979 Luminy paper ([Er79e], p. 74, where the sum is written $g(n)$) asks for an asymptotic formula and calls it easy to prove that $\frac1X\sum_{n\le X}g(n)\to\infty$; the 1982 survey says on p. 66 that he hopes to prove that $G(n)/\tau(n)$ has a distribution function, and a note on p. 67 adds that he and Tenenbaum proved in July 1981 that it has a continuous distribution function, a result on the ratio whose existence part the 1983 paper publishes as Théorème 1, beside the mean value. A Lean development of August 2026 in Boris Alexeev's repository proves $\sum_{n\le X}G(n)\sim X\log X$ and is linked from the Erdős--Tenenbaum claim page as a formalization of their result. The formal-conjectures statement file linked under Formalization points to that development as the formal proof of both parts, and the pull request that added it (#6383, merged 2026-09-20) records that its author rebuilt the development's closure against that repository's Mathlib, with #print axioms reporting only propext, Classical.choice and Quot.sound, and compiled a bridge from the development's definition of $G$ to the file's own statements. This corpus built the development at a pinned commit of 2026-09-15 and audited its statement, which the Erdős--Tenenbaum claim page records as formalized evidence beside the refereed paper; the build certifies the divergence and the leading term of the mean value, not the paper's error term. The site records Tao's suggestion that the problem was a slip that Erdős corrected a year later into Problem 448. The literature search behind this account, dated 2026-10-07, covered the site's page and discussion thread, the 1979 paper, the 1982 survey, Erdős and Tenenbaum's 1983 paper, Tenenbaum's 2013 survey, Alexeev's repository and the formal-conjectures file.

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_1982_my_favourite_problems_which_recently_have
• erdos_1979_unconventional_problems_number_theory_asterisque
• tenenbaum_2013_erdos_unconventional_problems_number_theory
• tenenbaum_2013_erdos_unconventional_problems_number_theory / estimate_p9
• weingartner_2015_practical_numbers_distribution_divisors
• weingartner_2015_practical_numbers_distribution_divisors / corollary_1
• weingartner_2015_practical_numbers_distribution_divisors / theorem_1