Many mountaineering agencies claim that there are 1310 peaks over 6000m in Nepal alone. Other agencies claim that there are thousands of such peaks in the world, meaning that no comprehensive list of peaks over 6000m probably exists. This makes sense, as peaks should intuitively follow some sort of power-law distribution, meaning that there should be much more peaks between 6000m and 7000m than between 7000m and 8000m.
Good sources are hard to find. A large part of the problem is that 6000m peaks are not really that significant in the Himalayas, where valleys can rise to over 5000m, and even passes can be over 6000m. To make the things worse, official Nepali records are generally not online. There is a list of 326 peaks in Nepal open for mountaineering, however, and most of the peaks seem to be over 6000m.