Problem #822: Does the set of integers of the form $n+\phi(n)$ have...

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Progress

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 $\omega$.