Problem #258: Let $a_1,a_2,\ldots$ be a sequence of integers with $a_n\to...

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Progress

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.