I would like to find a truly general formula for the OLS case, but a couple of observations: Start with $n$ to well-estimate the intercept, e.g., $n=100$, then add $k$ observations per estimated or entertained parameter. When the signal:noise ratio is low, simulations I have done point to $k=15$. I'd like to have a formula that takes into account the signal:noise ratio (e.g., true $R^2$). When this ratio is high you can estimate more parameters.
But we need to decide on the metric for model performance before answering the question. I don't think that power is the proper metric. What I prefer, and what the above is based on, is finding $n$ such that the apparent ordinary $R^2$ for the model is an unbiased estimate of the true long-run model performance.
I'm going to expand on this at Rules of thumb for minimum sample size for multiple regression
