Problem 258

Erdős Problem #258: declared status 'proved'. Formalized: yes. 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 $n$? Current best: Erd\H{o}s [Er48] proved that $\sum_n \frac{d(n)}{t^n}$ is irrational for any integer $t\geq 2$. Prize: no. Tags: irrationality.