And if I understand right, the next two bombs were used in japan and we were out of bombs for quite some time after that.
It took both bombs and another B-29 conventional bombing run (inadvertantly disrupted a coup to overthrow the emperor and keep fighting) to bring about the Japanese surrender.
Actually there was one more bomb being prepared at Los Alamos. Tibbets had already sent three B-29's back to the United States to pick it up as soon as it was ready.
We should have used #3 on Chinese on the Yalu River in 1950.
I believe that the U.S. had a production capacity of about 12 bombs a month in August 1945. And we did indeed produce at full capacity because Truman did not trust the Russians who out numbered us in Europe by about 4-1.
There actually was a third, though it was in pieces. As I recall the details, the core was in Utah, and the rest of the parts in California and Alamagordo...but it could have been slapped together (so to speak) and deployed during August.