Is there a result that connects the discriminant of an order to the discriminants of the generators of an integral basis for the order?

Let $$\left\{ 1 , \omega_1, \dots , \omega_{n-1} \right\}$$ be an integral basis for an order $$\mathcal{O}$$ of a number field of degree $$n$$ over $$\mathbb{Q}$$, let $$\Delta$$ be the discriminant of $$\mathcal{O}$$, and let $$D \left( \omega_j \right)$$ denote the discriminant of $$\omega_j$$. Is there a result that connects the discriminant of $$\mathcal{O}$$ to the discriminants of the generators of an integral basis for $$\mathcal{O}$$? e.i. something like $$$$\Delta = \gcd \left\{ D\left( \omega_j \right) \right\}_{j=1}^{n-1},$$$$ which would claim that the indices $$a_j$$ of $$\mathbb{Z}[\omega_j]$$ (for $$j \geq 1$$) in $$\mathcal{O}$$ are relatively prime? If the equation set out above is false in general, what is known about $$\gcd \left\{ a_j \right\}_{j=1}^{n-1}$$?