  We classify all Kutasov-Seiberg type dualities in large $N_c$ SQCD with
adjoints of rational $R$-charges. This is done by equating the superconformal
index of the electric and magnetic theories: the obtained equation has a
solution each time some product of cyclotomic polynomials has only positive
coefficients. In this way we easily reproduce without any reference to the
superpotential or the choice of the equations of motion (classical chiral ring)
all the known dualities from the literature, while adding to them a new family
with two adjoints with $R$ charges $\frac{2}{2k+1}$ and $\frac{2(k+1)}{2k+1}$
for all integers $k>1$. We argue that these new fixed points could be in their
appropriate conformal windows and in some range of the Yukawas involved a low
energy limit of the $D_{2k+2}$ fixed point. We try to clarify some issues
connected to the difference between classical and quantum chiral ring of this
new solution.
