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Claim. Let $\tau_\perp(n)$ be the function of Problem 1100 and $M(x)=\max_{n\le x}\tau_\perp(n)$. Samuel Korsky's manuscript Many Coprime Consecutive Divisors (dated 30 July 2026, 8 pages, the file linked above) proves (its Theorem 1.1) that

$$\liminf_{x\to\infty}\frac{\log M(x)\,\log\log x}{\log x} \ge c_*=\tfrac12\max_{0<\beta<1/2}\bigl(\sqrt{g(\beta)}-\sqrt{h(\beta)}\bigr)^2 =0.03648\ldots,$$

where $h$ is the binary entropy and $g(\beta)=-2\beta\log\beta-(1-2\beta)\log(1-2\beta)$, the maximum taken at $\beta=0.2368\ldots$; with the pointwise upper bound of Erdős and Tenenbaum, which gives $\limsup\le\log(3\cdot2^{-2/3})$, it concludes (its Corollary 1.2) that

$$M(x)=\exp\Bigl(\Theta\Bigl(\frac{\log x}{\log\log x}\Bigr)\Bigr),$$

and leaves the exact constant open. The construction takes $k$ primes from $(k^A,2k^A]$ with $A>2$ and lets $n$ be their product, so that $\log n\sim Ak\log k$; since all the primes have comparable size, the products of $\beta k$ of them lie at a common scale, and a pigeonhole over short bins yields many pairs of disjoint $\beta k$-subsets whose products are within a factor $1+e^{-\lambda k}$ of each other. A divisor lying between such a close coprime pair would force two independent near-equalities between products of at least $t$ primes, each of probability at most the reciprocal of a binomial coefficient by unique factorization, so a first moment shows that most close pairs are consecutive divisors; optimizing $\beta$, $A$ and $\lambda$ gives $c_*$.

Submission note. Posted to erdosproblems.com as a proof claim by Samuel Korsky (account SamKorsky) on 31 July 2026, giving "GPT 5.6-Pro" as the AI used:

Let $M(x):=\max_{n\le x}\tau_{\perp}(n).$ Answering (in the negative) the second question of Erdos (which should likely be weakened based on my literature review comment), we show that

$$> M(x)=\exp\!\left(\Theta\!\left(\frac{\log x}{\log\log x}\right)\right).$$

The

proof builds partly on ideas of Ross. We choose $k$ random primes from $(k^A,2k^A]$ and let $n$ be their product. Since the primes are comparable in size, divisor products supported on about $\beta k$ primes lie on similar scales, and a binning argument produces many very close pairs. The polynomial size of the primes keeps $n$ small enough for these pairs to give the desired maximal-order lower bound. Finally, any divisor lying between a close coprime pair forces two rare approximate equalities between prime products, so such obstructions are negligible. The rest is technical, making these estimates precise. Notes: The general idea to pick $k$ primes in $(k^A,2k^A]$ came from a joint discussion between GPT and myself, but all technical details, which are the bulk of the work here, were outsourced to GPT (and human-reviewed).

Covers. The maximal order of $\tau_\perp$ up to the constant in the exponent. This answers in the negative the question Erdős asked in [Er85], display (27), whether $\tau_\perp(n)<\exp(\epsilon\log n/\log\log n)$ for every $\epsilon>0$ and all large $n$, a question not in the site's statement; it also answers the site's second question in the negative, but that question was already refuted by the weaker bound $\exp(((\log2)^2+o(1))\log x/(\log\log x)^2)$ of the accepted Erdős--Tenenbaum page, which the manuscript cites as the earlier lower bound and which the author says on the thread should lead to a weakening of the question. The manuscript says nothing about the first question or about $g(k)$.

Depends on. [[problems/divisors/E1100/claims/1989_03_01_erdos_tenenbaum|Erdős and Tenenbaum's bounds]] supply the upper half of Corollary 1.2: the inequality $\tau_\perp(n)\le3\tau(n)B_1^{\omega_1(n)}$ of their Théorème 1 ((1.3) with $v=1$), where $B_1=3\cdot2^{-5/3}$ and $\omega_1(n)$ counts the primes dividing $n$ exactly once. It holds for every $n$ and gives the constant $\log(3\cdot2^{-2/3})$; the manuscript cites their Theorem 1 and, for the same constant, a general inequality of de la Bretèche. Their form $\tau_\perp(n)\ll\tau(n)^{1-c}$ is proved only for squarefree $n$, and the $\Theta$ upper half alone already follows from $\tau_\perp(n)<\tau(n)$. The lower bound is the manuscript's own.

Authorship and system. The author's acknowledgments say that the idea of taking $k$ primes from $(k^A,2k^A]$ arose in a discussion with GPT-5.6 Pro, that the system then assisted extensively with the derivations, estimates, exposition and revision, and that the author reviewed and verified the work and takes responsibility for it; the forum entry names the system as GPT 5.6-Pro and says the technical details were outsourced to it and human-reviewed. The manuscript credits the general strategy of seeking close coprime divisor pairs and removing those with an intervening divisor to Michael M. Ross's preprints on the squarefree problem, whose own claim is Ross's golden-ratio bound.

Standing. Posted on the problem's forum as a partial proof claim on 2026-07-31 with the manuscript as its external link. The claim carried one comment, by a forum commenter, noting that the idea of taking integers composed of exactly $k$ primes from an interval $(x,2x]$ was already used by Erdős and Hall in their lower bound. The site's curator has not commented, the problem's label is unchanged, the manuscript is not refereed and nobody has recorded accepting it, so the claim is claimed.