Wouldn't he be the German Emperor Frederick I? And if it was for Prussia, shouldn't it have been Frederick V?
Supposedly Frederick III preferred to be called Frederick IV when he ascended to the throne, considering himself to be a successor to the Holy Roman Emperors, however that seems to have been politically unviable (this is not a problem his father had because he was the first King of Prussia called Wilhelm and there were no Holy Roman Emperors with that name either). So he kept the same regnal name he would have had as the King of Prussia. There were only two Kings of Prussia named (just) Frederick/Friedrich:
There were four Prussian kings called Frederick William but they were counted separately. So I'm not sure where did you get the V from.