Approximately how many people lived in the Louisiana Purchase when it was bought by the US? What were the largest towns in the Purchase at that time, besides New Orleans?

Okay, I'll preface this by saying the last time I did any research into colonial Louisiana was in 2013. I do the history of Arkansas, but my research over the past two years has been on the period between 1870 through 1930. So with that caveat..

As you probably know, Louisiana didn't receive as many European migrants as other colonies did. This is for a couple of different reasons -

First, the French viewed their colonies differently than other European colonial powers. They weren't so much thought of as places for Frenchmen to live, but as places for France to make a profit.

Second, France had a massively profitable colony (San Domingue/Haiti) that drew the vast majority of French settlement and attention. Haiti is almost ideal for growing sugarcane, which brought massive profits to French plantation owners and the French state. Louisiana is also good for growing sugarcane, but it never received the same attention as Haiti did. Merchant ships stopped there less often, the price of production is higher, and so on.

Thid, there was no knowledge of any easily exploitable mineral resources. There were gold deposits (relatively) nearby the upper Mississippi river, but that area was largely unexplored. The Ozark and Ouachita mountains definitely don't have gold and were controlled by the Osage, which were definitely hostile to the French.

So, for these reasons, the major economic activity of French Louisiana was the fur trade. Trading fur doesn't require a large population, nor does it necessarily attract one. Most of the French fur trade was conducted by purchasing pelts from Native Americans.

I don't recall coming across any hard figures of how many Europeans settled in Louisiana prior its purchase by the United States. My gut feeling says that less than ten thousand Europeans (mostly Frenchmen, a colony of Germans, and a scattering of Spaniards) settled in the entire Louisiana territory. The books that I read for my research will imply about that number. And, for what it's worth, Wikipedia claims that around 7,000 Europeans migrated. Either way, French Louisiana was definitely not a settler colony.

The only major settlements outside of modern-day Louisiana were Arkansas Post and Saint Louis/Cahokia. Both settlements are at major river intersections, and functioned more as trading posts than as true "settlements." The population of each settlement fluctuated, but Arkansas Post probably had about a hundred Europeans at most. The Encyclopedia of Arkansas doesn't mention a population count for Arkansas Post, but because Arkansas Post is literally in the middle of a swamp, it never became particularly populated. The confluence of the Missouri and Ohio Rivers with the Mississippi is obviously more habitable. Despite the advantage of not being in a swamp, the population of Saint Louis maybe topped out at five hundred. Transport from New Orleans to Arkansas or Missouri was laborious, which further limited settlement (I certainly wouldn't want to paddle UP the Mississippi River!)

Estimating the Native American population of the region is even more difficult, for obvious reasons. This is a completely unsupported assertion, but by the Louisiana Purchase, there were probably about 50,000 to 75,000 Native Americans living in the region.

As I said before, it's been a good minute since I've done any research on French Louisiana, but I remember these books being helpful.

Giraud, Marcel. A History of French Louisiana. Translated by Joseph C. Lambert. Vol. 1. Baton Rouge: Louisiana State University Press, 1974. Eccles, William J. The French in North America, 1500-1765. East Lansing: Michigan State University Press, 1998. Allain, Mathé. “Not Worth a Straw”: French Colonial Policy and the Early Years of Louisiana. Lafayette, LA.: University of Southwestern Louisiana, 1988. Aron, Stephen. American Confluence: The Missouri Frontier from Borderland to Border State. Indiana University Press, 2009.