There are plenty of variables that influence "productivity" (however that is defined) but I imagine that the amount of sleep required for maximum productivity varies from person to person.

In my personal experience I've known people who require 3 to 4 hrs/night, and others (myself included) who need between 8 and 9. During times when I'm feeling my absolute best, physically and mentally, I get about 2 hours of rigorous exercise, 8 to 10 hours of productive work, and roughly 8 hours of sleep a day with the rest of the time filled in by preparing and eating food, taking short breaks, hobbies, relaxing, etc.

Wild hypothesis: Most people would agree that the amount of sleep they require to "feel and do their best" (assuming ideal exercise, diet, etc.) falls within a normal distribution with mean 7 hours and standard deviation of +/- 3 hours.

--- edit: spelling