For a quasinilpotent operator T on a Banach space X, Douglas and Yang defined kx=limsupz→0ln‖(z−T)−1x‖ln‖(z−T)−1‖ for each nonzero vector x∈X, and call Λ(T)={kx:x≠0} the power set of T. They proved that the power set have a close link with T's lattice of hyperinvariant subspaces. In this paper, we compute the power set of some weighted shifts on ℓp for 1≤p<∞. The following results are obtained: (1) If T is an injective quasinilpotent forward unilateral weighted shift on ℓp(N), then Λ(T)={1} when ke0=1, where {en}n=0∞ be the canonical basis for ℓp(N); (2) There is a class of backward unilateral weighted shifts on ℓp(N) whose power set is [0,1]; (3) There exists a bilateral weighted shift on ℓp(Z) with power set [12,1].