OPEN
This is open, and cannot be resolved with a finite computation.
Let $a_1,a_2,\ldots$ be a sequence of integers with $a_n\to \infty$. Is\[\sum_{n} \frac{\tau(n)}{a_1\cdots a_n}\]irrational, where $\tau(n)$ is the number of divisors of $n$?
Erdős and Straus
[ErSt71] proved this is true if $a_n$ is monotone, i.e. $a_{n-1}\leq a_n$ for all $n$. Erdős
[Er48] proved that $\sum_n \frac{d(n)}{t^n}$ is irrational for any integer $t\geq 2$.
Erdős and Straus further conjectured that if $a_{n-1}\leq a_n$ for all $n$ then\[\sum_{n} \frac{\phi(n)}{a_1\cdots a_n}\]and\[\sum_{n} \frac{\sigma(n)}{a_1\cdots a_n}\]are both irrational.
This problem has been
formalised in Lean as part of the
Google DeepMind Formal Conjectures project.
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This page was last edited 28 September 2025.
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 #258, https://www.erdosproblems.com/258, accessed 2025-11-16
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