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arXiv · 2506.04883

On the number of divisors of Mersenne numbers

Vjekoslav Kovač·Florian Luca
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Abstract

Denote $f(n):=\sum_{1\le k\le n} \tau(2^k-1)$, where $\tau$ is the number of divisors function. Motivated by a question of Paul Erd\H{o}s, we show that the sequence of ratios $f(2n)/f(n)$ is unbounded. We also present conditional results on the divergence of this sequence to infinity. Finally, we test numerically both the conjecture $f(2n)/f(n)\to\infty$ and our sufficient conditions for it to hold.

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Vjekoslav Kovač, Florian Luca. 2025-06-05. On the number of divisors of Mersenne numbers. https://arxiv.org/abs/2506.04883

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