Is it true that polynomials of the form :

$ f(x)= x^n+x^{n-1}+...+x^{k+1}+ax^k+ax^{k-1}+...a$

where $gcd(n+1,k+1)=1$ and $ a\in \mathbb{Z^{+}} $

are irreducible over ring $\mathbb{Z} $ of integers ?

Neither of Eisenstein's criterion and Cohn's criterion cannot be applied on the polynomials of this form. I have checked a lot of cases and it seems to be true.