Problem 448

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claims/: The 2 claim pages of Problem 448, one per claimant's result; the problem's standing derives from them.

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Statement. Let $\tau(n)$ count the divisors of $n$ and $\tau^+(n)$ count the number of $k$ such that $n$ has a divisor in $[2^k,2^{k+1})$. Is it true that, for all $\epsilon>0$,

$$\tau^+(n) < \epsilon \tau(n)$$

for almost all $n$?

Status. DISPROVED (LEAN). The site's label; Erdős and Tenenbaum showed in 1981 that the integers with $\tau^+(n)<\epsilon\tau(n)$ do not have density one for small $\epsilon$, and the Lean is a third-party formalization of their disproof, not built here, as the claim page below records.

Source. erdosproblems.com/448, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #448, https://www.erdosproblems.com/448.

References.

• [ErTe81] Erdős, P. and Tenenbaum, G., Sur la structure de la suite des diviseurs d'un entier. Ann. Inst. Fourier (Grenoble) (1981), ix, 17-37.
• [Fo08] Ford, Kevin, The distribution of integers with a divisor in a given interval. Ann. of Math. (2) (2008), 367-433.
• [HaTe88] Hall, Richard R. and Tenenbaum, Gérald, Divisors. (1988), xvi+167.

Formalization. Statement in formal-conjectures, read at its commit of 2026-09-18: erdos_448 with the answer False and no proof, marked solved and pointing for its formal proof to the Lean file in Boris Alexeev's repository that the claim page below links at its pinned commit; neither file has been built here.

Current assessment

The question is the site's formulation, accessed: whether, for every $\epsilon>0$, almost all $n$ satisfy $\tau^+(n)<\epsilon\tau(n)$, where $\tau^+(n)$ counts the dyadic intervals $[2^k,2^{k+1})$ holding a divisor of $n$. The answer is no.

Erdős and Tenenbaum [ErTe81], Théorème 1, bound the upper density of $\{n:\tau^+(n)\le\alpha\tau(n)\}$ by $c(\varepsilon)\alpha^{1-\varepsilon}$ for every $\varepsilon>0$, which is below one for small $\alpha$, so the exceptional set keeps positive lower density and the conjectured statement fails (card). The site's commentary records the order of that upper density as $\alpha^{1-o(1)}$ and the sharper bound $\ll\alpha\log(2/\alpha)$ of Hall and Tenenbaum [HaTe88], Section 4.6, who also prove that $\tau^+(n)/\tau(n)$ has a distribution function; that theorem settles the question on its own and is recorded as a pending claim on [[problems/divisors/E0448/claims/1988_09_15_hall_tenenbaum|Hall and Tenenbaum 1988]], since the book is a monograph and the site credits the disproof to Erdős and Tenenbaum. The claim page [[problems/divisors/E0448/claims/1981_01_01_erdos_tenenbaum|Erdős and Tenenbaum 1981]] records the theorem, the refereed venue, the curator's credit and the Lean formalization, and the problem's standing derives from it.

The Lean behind the site's qualifier is Boris Alexeev's formalization of the Erdős–Tenenbaum disproof, with Codex and GPT-5.6 Sol named as its formal authors, which proves the negation of the formal-conjectures statement erdos_448; the formal-conjectures file itself states the answer without proof. Neither has been built or audited in this repository, so the standing rests on the refereed paper and the curator's credit, not on a kernel check made here.

Erdős and Graham's companion question, a good estimate for $\sum_{n\le x}\tau^+(n)$, is not part of the statement; Ford [Fo08] answered it with $\sum_{n\le x}\tau^+(n)\asymp x(\log x)^{1-\alpha}(\log\log x)^{-3/2}$, $\alpha=1-(1+\log\log2)/\log2=0.08607\ldots$. The density version with a single interval $(n,2n)$ is Problem 446, and Problem 449 is a neighbor. The account rests on the site page, the Erdős–Tenenbaum card, the two Lean files and the journal record.

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_1978_unconventional_problems_divisors_integers
• erdos_1981_sur_la_structure_de_la_suite
• erdos_1981_sur_la_structure_de_la_suite / theorem_1
• ford_2008_distribution_integers_divisor_given_interval
• ford_2008_distribution_integers_divisor_given_interval / corollary_5
• tenenbaum_2013_erdos_unconventional_problems_number_theory
• tenenbaum_2013_erdos_unconventional_problems_number_theory / equation_15