mp = 1.6726231*10^(-27); me = 9.1093897*10^(-31); 
    Z = 1; h = 6.626076*10^(-34); ϵ0 = 
      8.854188*10^(-12); 
    e = 1.602177*10^(-19); μ = me*mp/(me + mp);

NSolve[(n^4 + 2 n^3 - 3 n^2)/(4 (2 n + 1)) - 1 - 
        5*10^5]

n = 158;

(Z*e^2/4/π/ϵ0)^2*μ/2/h^2/
              4/π^2/n^2 // N

5.599649647608985`*^-26/e