Disjoint supercyclic powers of weighted shifts on weighted sequence spaces

We characterize the disjoint supercyclicity of finitely many different powers of weighted shifts acting on the weighted sequence spaces l^2(N,w), c_0(N,w), and l^2(Z,w), c_0(Z,w), where w=(w_i)_i is a positive weight sequence satisfying w_i geq 1 for every iin N (or iin Z).