There aren't really brushes made specifically for applying a primer. For one thing, I think far more primers are applied by spraying than by hand-brushing. But when I have applied a primer coat by hand, I use one of the cheaper brushes that I keep for utility work like that. I bought a bag of brushes made in Korea by a company called Aristo-Craft, for example. They have plastic handles and synthetic fiber bristles. I use a #6 for things like this-the bristles are a half-inch long, and about a quarter-inch wide.
But more important to ask is, what are you using as your primer, and how do you clean your brushes? If you're using an acrylic for your primer, you can get away with using water or isopropyl to clean, but paint reside will still build up in the ferrule. So every so often, I'd give the brush a cleaning with lacquer thinner, which should remove the residue. If you're priming with an enamel, you should be able to use mineral spirits or turpentine/turpenoid to clean. And if you're using a lacquer, then you'd need to use lacquer thinner.
I use the same method to clean, though, regardless of the chemical makeup of my paint. I dip the head of the brush in a jar of the appropriate thinner and stir it around a little, till I see the paint dissolving into the solution. I remove the brush and then drag the head of the brush gently across a piece of paper towel, repeating as necessary till there is no color coming off the brush onto the paper towel. I re-shape the points and stand the brushes to air-dry.
Of course, depending on how cheap your utility brushes are, you might just throw them out after a couple of uses.
Hope that helps, prosit!
Brad