Prove that $|n-\frac{\sigma_0(n)\phi(n)}{2}|$ equal to a prime implies that n is a semiprime

Question

Unfortunately I don't have much to add. The statement appears to be true based on examination of small n. Menon's identity gives an alternative respresentation of $\sigma_0(n)\phi(n)$, but I don't know if this is a potential path towards proof.

Answer 1

It is not true. Consider $n=255=3 \cdot 5 \cdot 17$. We have $\sigma_0(n) = 8$ and $\phi(n) = (3-1) \cdot (5 - 1) \cdot (17 - 1) = 128$, so $\vert n - \frac{\sigma_0(n) \phi(n)}{2}\vert = 257$ which is prime. But $255$ is not a semiprime.