I think you're confusing '1 week in the primary' (which is JUST an average estimate) with '3 weeks in the bottle' to condition and carb the beer.
as orfy correctly states, it should really take the same amount of time unless you pitch an ungodly amount of yeast, which can cause off flavors if you go too far overboard.
Primary: English Mild
On tap: Pale Ale, Lancelot's Wheat, English Brown Ale, Steam Beer, HoovNuts IPA
Bottled: MOAM, Braggot, Raspberry Melomel, Merlot, Apfelwein, Pyment, Sweet mead, Cabernet
Gal in 2009: 27, Gal in 2010: 34, Gal in 2011: 13, Gal in 2012: 10