With my first daughter I was 40 weeks + 1 day. My OB had unnecessarily suggested (and put into action) an induction 4 days prior to that. The induction technically failed, but the stress likely helped along my real labor. Of course, after days of fruitless induced labor and the after-pains from that, my "real" 33-hour labor was a very rough experience.
My second daughter was born at 40 weeks + 5 days. I had allowed my midwife to sweep my membranes a couple days prior. I'm not sure if that played any part in my timing, though. In fact, I think it may have just been responsible for the miserable 24 hours of prodromal labor that I experienced before my super fast "real" labor kicked in. Thankfully, we were intervention free aside from my consent to the membrane sweeping (which was likely based in fear of having to transfer to a hospital for another real induction if my daughter didn't come out soon enough.) The end results were 100% different from my induced hospital birth.
Next time I'm going to ignore all suggestions about speeding things along. My body knows what it's doing, and apparently it just likes to cook babies to perfection.