Problem 893

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claims/: The 2 claim pages of Problem 893, one per claimant's result; the problem's standing derives from them.

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Statement. If $\tau(n)$ counts the divisors of $n$ then let

$$f(n)=\sum_{1\leq k\leq n}\tau(2^k-1).$$

Does $f(2n)/f(n)$ tend to a limit?

Formulation. The question whether $f(2n)/f(n)$ tends to a limit is read, as the site and Kovač and Luca read it, as allowing the limit $+\infty$. After Kovač and Luca proved the ratios unbounded, the site kept the problem OPEN, and they describe what remains as deciding whether $f(2n)/f(n)\to\infty$ or the ratio has no limit. The formal-conjectures statement, as last changed on 18 September 2026, likewise asks whether the ratio tends to infinity. Read as asking for a finite limit, the question is answered no by their Theorem 1.

Status. The site labels the problem OPEN. Every finite limit is ruled out by Kovač and Luca's unboundedness theorem; divergence to infinity is proved only under unproven conjectures, on their conditional page.

Source. erdosproblems.com/893, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #893, https://www.erdosproblems.com/893.

References.

• [Er98] Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.
• [KoLu25] V. Kovač and F. Luca, On the number of divisors of Mersenne numbers. arXiv:2506.04883 (2025); published in Experimental Mathematics (online 20 May 2026), doi:10.1080/10586458.2026.2636060.

Formalization. Statement in formal-conjectures.

Current assessment

The site labels the problem OPEN. Under the formulation above, the open question is whether $f(2n)/f(n)\to\infty$. Kovač and Luca (card) prove the ratios unbounded, which rules out every finite limit (claim page). They also derive $f(2n)/f(n)\to\infty$ from either of two unproven conjectures (conditional claim page), and their computations support divergence. Erdős [Er98] expected no simple asymptotic formula for $f(n)$, since it grows too fast.

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.

• kovac_2025_number_divisors_mersenne_numbers
• kovac_2025_number_divisors_mersenne_numbers / proposition_2
• kovac_2025_number_divisors_mersenne_numbers / theorem_1
• kovac_2025_number_divisors_mersenne_numbers / theorem_3