Erdős Problem #258 — Let $a_1,a_2,\ldots$ be a sequence of positive 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 Problem #258 asks whether, for every positive-integer sequence with $a_n\to\infty$, the series $\sum_{n\ge 1}\tau(n)/(a_1\cdots a_n)$ must be irrational.

Known results

• Erdős proved the fixed-base case $\sum_n\tau(n)/t^n$ irrational for every integer $t\ge 2$ (1948).
• Erdős and Straus proved the result when $a_{n-1}\leq a_n$ (1971).

Affirmative solution claim

Chojecki and GPT-5.4 Pro are reported to have proved the unrestricted case, using Tao and Teräväinen’s 2025 theorem on prime factors of consecutive integers. A note states the full theorem, and Lean formalizations record the result, but the accessible formalization contains sorry declarations and no independently verified completed proof was found.

Current status (as of June 2026): The unrestricted statement is claimed solved affirmatively, while independent verification of the proof remains outstanding.