An example for $n=7$ is given in  J. W. S. Cassels, Factorization of polynomials in several variables, Proc. Fifteenth Scandinavian Congress (Oslo, 1968), vol. 118, Lecture Notes in Mathematics, Springer, Berlin, pp. 1-17. This reference is from https://oeis.org/A112090 and is quite technical.

here I type the example from the paper in a sort of computer-readable format

    l=(1+sqrt(-7))/2
    m=(1-sqrt(-7))/2
    # t a nonzero(?) parameter
    f(x)-g(y)=(x^3+l*x^3*y-m*x*y^2-y^3-(3*l+2)*t*x+(3*m+2)*t*y+t)*
              (x^4-l*x^3*y-x^2*y^2-m*x*y^3+y^4+2*(m-l)*t*x^2-
               7*t*x*y+2*(l-m)*t*y^2+(3-l)*t*x-(3-m)*t*y-7*t^2)
Sage script (with input f and (-)g taken from the paper

    z=QQ['z'].0
    K.<s>=NumberField(z^2+7)
    l=(1+s)/2
    m=(1-s)/2
    R.<x,y,t>=K[]
    # t a nonzero(?) parameter
    f=x^7-7*l*t*x^5+(4-l)*t*x^4+(14*l-35)*t^2*x^3-(8*l+10)*t^2*x^2+(3-l+7*(3*l+2)*t)*t^2*x
    g=-y^7+7*m*t*y^5+(4-m)*t*y^4-(14*m-35)*t^2*y^3-(8*m+10)*t^2*y^2-(3-m+7*(3*m+2)*t)*t^2*y-7*t^3
    (f+g).factor()

outputs

    (x^3+(1/2*s+1/2)*x^2*y+(1/2*s-1/2)*x*y^2-y^3+(-3/2*s-7/2)*x*t  + (-3/2*s + 7/2)*y*t + t) * 
    (x^4 + (-1/2*s - 1/2)*x^3*y - x^2*y^2 + (1/2*s - 1/2)*x*y^3 + y^4 + 
     (-2*s)*x^2*t - 7*x*y*t + (2*s)*y^2*t + (-1/2*s + 5/2)*x*t + (-1/2*s - 5/2)*y*t - 7*t^2)

which is the factorisation that should be like the one above.
You are welcome to spot a typo, if any.