Given an odd integer in $\mathbb Z_{\geq0}$ of $n$ bits let $a_{n-1}a_{n-2}\dots a_1a_0$ be its binary expansion where $a_{n-1}=a_0=1$.

Call an $n$ bit prime $f(n)$-robust if we flip any $f(n)$ of its $n-2$ bits $a_{n-2}\dots a_1$ the resulting $n$ bit odd number is not prime.

1. Do we know how large $f(n)$ can be at least under any reasonable conjectures?

I think $f(n)=\Omega(\log n)$ might be known but $f(n)=\omega(\log n)$ might be possible.

2. Do we know at least average case behavior of $f(n)$?