The short answer is f/4 gathers half of the light as f/2.8, f/2.8 gathers half of the light as f/2, and f/2 gathers half of the light as f/1.4.

Lenses manufacturers want to make it so since the shutter speeds are also half of each other per step (1/1000, 1/500, 1/250, etc).

This makes it easy in the film era where one has to determine an exposure level first, then choose the aperture for each shot. Say according to my light meter f/2.8 paired with 1/250 (with given ISO on the film, of course) gives me the right exposure, then if I now want a larger aperture f/2 for my portrait, I would just set the shutter speed to be 1/500 to balance off the excess light.

If you look at this vintage rangefinder, you would actually see the silver metal rings, one for aperture and one for shutter speed, are sticked together, so that it is easier for the user to change both of them by the same amount.

For stops between these common aperture levels, e.g f/2.2, f/2.4, f/3.5 etc, my guess is the largest aperture is limited by the physical size of the lenses. A large aperture brings more difficulty to the design hence lenses with large apertures are usually larger in size. Say the manufacturer wants to produce a small lens for portrait, they would be satisfied with something larger in size than a f/4, but something with f/2.8 is too large for them. So they pick some aperture level that gives them the right size, say f/3.5.

Of course all of this wouldn't matter today because the light meter can essentially determine the exposure at any given aperture size. The stops are still there for the above historical reasons.