We typically go to mountains to get away from it all, especially other humans. I'd always wanted to investigate the relationship between population density and elevation. Surprise! Such a relationship exists. The higher you go, the lower the density; in fact the relationship is even stronger than exponential. Even more interesting than that is understanding why. But first: figure below is from Cohen and Small 1998:
Note the bump at 2,300 m and again at 4,000+. 2,300 is mostly people on the Mexican Plateau, and 4,000 is Tibet + the Andes.
I plugged in data just for ~250 cities in the state of Utah, and got a worse R^2 than I saw for Cohen and Small's data (0.25 vs >0.9 for theirs), but this isnt' that granular; when I tried to do the same thing using mean elevations and population densities for the 50 states it was a mess. I think it would work a lot better at the county level.
So why is this? You might be tempted to speculate that it has to do with the limits of human physiology. That is to say, the higher you go, the more uncomfortable people are (thin air, cold) and the less adaptable, right? It's hard to imagine (for example) that Las Vegas at 665 meters would be a more comfortable place at a lower elevation. And physiologically, as it turns out, in the second half of the twentieth century medical anthropologists studied people in Tibet and the Andes extensively for their physiological adaptations to altitude - the underlying mutations for which have now been characterized (and in the Tibetan case have all occurred in the last 3,000 years!). But the idea that a few hundred meters of elevation will start impacting physiology and population growth falls apart both in terms of common sense and as a direct implication of other work. Any decrease in fertility caused by altitude would cut right at population growth - but this turns out not to be a big concern even for Himalayans living much higher.
You could also argue that the world's large cities tend to be seaports, which are at low elevation, so it's ease of transportation that gives us this bias toward lower elevations; but this is more true for New World that got settled by the sea than the Old World that got settled by land, and people must be pretty lazy if what's keeping them at sea level is their ancestors having gotten off a boat there a few centuries ago.
Terrace farming in the Himalayas.
I bet they would rather just have flat fields.
Beyond some historical accidents, the answer is likely to be mostly "agriculture". The overall population distribution we see in the world today basically reflects how early people in a certain part of the world adopted agriculture and how effective it was, given the crop and the climate, and escaped the Malthusian cycle; hence the highest densities being in a band running from east, southeast, and south Asia. The Middle East started early and although the marginal environment was also a driver for state formation, it was still marginal, and the Fertile Crescent just can't compete with the Ganges or the Pearl River. (So it can be accurately said that on average, humans are Asian; hence this map). So the relationship between elevation and population is really about where agriculture is better, and it's better at lower elevations for many reasons. Otherwise it's hard to understand the Mexican bump at 2300 meters, where (guess what) agriculture was first invented in the New World.
So, if someone ever thinks of a way to do agriculture as easily in the mountains as on flat lands, the days of undeveloped remote mountains are over.
Cohen JE and Small C. Hypsographic demography: The distribution of human population by altitude. Proc. Natl. Acad. Sci. USA, Vol. 95, pp. 14009–14014, November 1998.
Goldstein MC, Tsarong P, Beall CM. High Altitude Hypoxia, Culture, and Human Fecundity/Fertility: A Comparative Study. American Anthropologist 85(1), March 1983.