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 (using topology of Riemann surfaces).

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.<l>=NumberField(z^2-z+2)
m=l.conjugate()
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 + (l)*x^2*y + (l - 1)*x*y^2 - y^3 + (-3*l - 2)*x*t + (-3*l + 5)*y*t + t) *
(x^4 + (-l)*x^3*y - x^2*y^2 + (l - 1)*x*y^3 + y^4 + (-4*l+2)*x^2*t-
7*x*y*t + (4*l-2)*y^2*t + (-l + 3)*x*t + (-l-2)*y*t - 7*t^2)
```

which is the factorisation that should be like the one above.
But it is not - there is a typo above (and in the paper: in the 1st factor the monomial l*x^3*y should be l*x^2*y. After this change everything checks out.