PROVED
This has been solved in the affirmative.
Does the set of integers of the form $n+\phi(n)$ have positive (lower) density?
A similar question can be asked for $n+\sigma(n)$, where $\sigma$ is the sum of divisors function.
This is true, and was proved by Gabdullin, Iudelevich, and Luca
[GIL24]. They also proved this with $\phi$ replaced by the divisor function $\tau$ or the prime-counting function $\omega$.
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This page was last edited 14 October 2025. View history
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Additional thanks to: msellke
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 #822, https://www.erdosproblems.com/822, accessed 2026-04-11
