%4 is huge when measuring the wavelength. Circa 1888 was the initial formula. It makes things ‘not fit’.
The Rydberg constant http://scienceworld.wolfram.com/physics/RydbergConstant.html has nothing to do with the radius of the proton. It is calculated from the electron mass, the proton mass, the electron charge, the speed of light, the permittivity of free space, and the Planck´s constant,