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We prove that a finite set of natural numbers $J$ satisfies that $J\cup\{0\}$is not Sidon if and only if for any operator $T$, the disjoint hypercyclicityof $\{T^j:j\in J\}$ implies that $T$ is weakly mixing. As an application weshow the existence of a non weakly mixing operator $T$ such that $T\oplusT^2\ldots \oplus T^n$ is hypercyclic for every $n$.

Disjoint hypercyclicity, Sidon sets and weakly mixing operators