The quantum number n, which describes the eigenvalue and the eigenfunction of a system, arises through solving the Schrodinger equation, which is eigen value problem.
The type of the equation differs depending on the systems. For example, in the case of an electron in a central force field, the radial part of the Schrodinger equation for the electron becomes the spherical Bessel differential equation, and in the case of an electron in a triangular potential well, the r Schrodinger equation for the electron becomes Airy differential equation. The solution of the spherical Bessel differential equation is the spherical Bessel function, and the solution of the Airy differential equation is the Airy function. t, and thus the form of the solution differs. The solution contains the quantum number in different ways depending on the the type of the differential equation. Because of this, the range of the value of quantum number n is defined by the nature of the solution function.
Mathematically, it is possible to introduce a newly defined quantum number n' through change of variables, such as "n' = n+1" or "n' = n+5" or whatever. But, the quantum number n often relate with