OPEN
This is open, and cannot be resolved with a finite computation.
Is\[\sum_{n\geq 2}\frac{\omega(n)}{2^n}\]irrational? (Here $\omega(n)$ counts the number of distinct prime divisors of $n$.)
Erdős
[Er48] proved that $\sum_n \frac{d(n)}{2^n}$ is irrational, where $d(n)$ is the divisor function.
Pratt
[Pr24] has proved this is irrational, conditional on a uniform version of the prime $k$-tuples conjecture.
Tao has observed that this is a special case of
[257], since\[\sum_{n\geq 2}\frac{\omega(n)}{2^n}=\sum_p \frac{1}{2^p-1}.\]
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This page was last edited 28 September 2025.
Additional thanks to: Vjekoslav Kovac and Terence Tao
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 #69, https://www.erdosproblems.com/69, accessed 2025-11-16
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