---

Claim. V. Kovač and F. Luca, On the number of divisors of Mersenne numbers, Experimental Mathematics (online 20 May 2026; card), Theorem 1 (numbered as in arXiv v4; Corollary 1 in v1): $\limsup_{n\to\infty}f(2n)/f(n)=\infty$. The proof goes through Proposition 2: for $f'(n)=\sum_{k\le n}2^{\tau(k)}$, $f'(2n)/f'(n)\to\infty$, proved with highly composite numbers. Primitive prime divisors give $\tau(2^k-1)\ge2^{\tau(k)}/4$, so $f\ge f'/4$. Hence $f(2n)/f(n)$ converges to no real number, which answers Problem 893 no if a finite limit is meant.

Covers. Every finite limit is ruled out. Whether $f(2n)/f(n)\to\infty$ stays open; the paper's conditional Theorem 3 is on its own page.

Depends on. No page of this wiki.

Acceptance. Refereed: Experimental Mathematics. The site credits the result but labels the problem OPEN, so its commentary is not acceptance.