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For a quasinilpotent operator $T$ on a Banach space $X$, Douglas and Yangdefined $k_x=\limsup\limits_{z\rightarrow0}\frac{\ln\|(z-T)^{-1}x\|}{\ln\|(z-T)^{-1}\|}$ for each nonzero vector $x\inX$, and call $\Lambda(T)=\{k_x: x\ne 0\}$ the power set of $T$. They provedthat the power set have a close link with $T$'s lattice of hyperinvariantsubspaces. This paper computes the power set of quasinilpotent weighted shiftson $l^p$ for $1\leq p< \infty$. We obtain the following results:  (1) If $T$ is an injective quasinilpotent forward unilateral weighted shifton $l^p(\mathbb{N})$, then $\Lambda(T)=\{1\}$ when $k_{e_0}=1$, where$\{e_n\}_{n=0}^{\infty}$ be the canonical basis for $l^p(\mathbb{N})$;  (2) There is a class of backward unilateral weighted shifts on$l^p(\mathbb{N})$ whose power set is $[0,1]$;  (3) There exists a bilateral weighted shift on $l^p(\mathbb{Z})$ with powerset $[\frac{1}{2},1]$.

Power Set of Some Quasinilpotent Weighted shifts on $l^p$