PROVED
This has been solved in the affirmative.
Let $\tau(n)$ count the number of divisors of $n$. Is the sequence\[\frac{\tau(n+1)}{\tau(n)}\]everywhere dense in $(0,\infty)$?
This follows easily from the generalised prime $k$-tuple conjecture. Eberhard
[Eb25] has proved this unconditionally, and in fact proved that all positive rationals can be written as such a ratio.
See also
[946].
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #964, https://www.erdosproblems.com/964, accessed 2025-11-16
