By the "easy" direction of Duffin-Schaeffer conjecture, it is known that if (*)$\sum_{q=1}^{\infty}\phi(q)f(q) < \infty $ (when $\phi(q)$ is euler totient function) then almost all numbers are not well approximated. When $\sum_{q=1}^{\infty}qf(q) < \infty$ it is the same as in Khintchine theorem.
Is there a function $\psi(q) = qf(q)$ such that $\sum_{q=1}^{\infty}\psi(q) = \infty$ and still (*) holds?
I have thought of defining it by something like $\psi(q) = \frac{1}{q}$ if $q$ prime and 0 else. But then, still, (*) doesn't hold. Does it require fine tuning or am I in a wrong direction?
Thanks in advance.