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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 $\omega$.
2025-10-20 00:00:00
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 $\omega$.
