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Let $a_n\to \infty$. Is \[\sum_{n} \frac{d(n)}{a_1\cdots a_n}\] irrational, where $d(n)$ is the number of divisors of $n$?
#258
:
[ErGr80]
irrationality
Erdős and Straus
[ErSt71]
have 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 for any integer $t\geq 2$.
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