6 cats in a household is fine! You are giving all these cats a home, which is much better than the alternative.
This type of hair loss is most commonly due to behavioral barbarism, food or environmental allergies, or, mites or fungal skin infections, such as ringworm. If it was due to an a fungal or mite infection other cats in your household would probably be exhibiting symptoms.
Please take him to the vet for evaluation. The vet can perform a skin scraping and other diagnostic tests. If it is truly behavioral, there are excellent behavioral modification medications that should help. If it is allergies, prednisolone, Atopica, and/or a special hypoallergenic diet may help. The significance of the lamp may be due to the fact that he feels safe away from the other cats on your desk, or he requires the heat from the lamp.
There is a pecking order in any multi-cat household and if he is lower on the pecking order there may be an alpha cat who is bullying him. Since he is too fearful to retaliate he pulls out his hair to try to relieve stress, and uses avoidance behavior. Bully's try to control and "own" areas of importance, such as litter boxes, food and water bowls and owners laps, favorite sleeping areas, etc. Please use one litter pan for each cat, plus one additional per household, to avoid additional behavioral problems
Thank you for the advice, I will make an appointment with our vet immediately.