Define the Erdős polynomial to be $f_n(x):= \sum \limits_{0 \leq i,j \leq n}^{}{x^{ij}}$ (the name is motivated by http://oeis.org/A027424).

For example for $n=5$, the polynomial is given by $x^{25}+2x^{20}+x^{16}+2x^{15}+2x^{12}+2x^{10}+x^9+2x^8+2x^6+2x^5+3x^4+2x^3+2x^2+x+11$.

Question 1: Is $f_n(x)$ always irreducible (over $\mathbb{Q}$)? If not, when is it irreducible?

It is irreducible for $n \leq 20$.

More generally, define for $k \geq 2$ the polynomial $f_n^k(x):= \sum \limits_{0 \leq i_1,i_2,...,i_k \leq n}^{}{x^{i_1 ... i_k}}$.

Question 2: Is $f_n^k(x)$ always irreducible? If not, when is it irreducible?

For $k=3$, it is irreducible for $n \leq 8$.