Given a continuous linear operator $T\in L(X)$ defined on a separable $\mathcal{F}$-space $X$, we will show that $T$ satisfies the Hypercyclicity Criterion if and only if for any strictly increasing sequence of positive integers   $\{ n_k\}_k$ such that $\sup_k\{ n_{k+1}-n_k\} < \infty $, the sequence $\{ T^{n_k}\}_k$ is hypercyclic. In contrast we will also prove that, for any hypercyclic vector $x \in X$ of $T$, there exists a strictly increasing sequence $\{ n_k\}_k$ such that $\sup_k\{n_{k+1}-n_k\}= 2$ and $\{ T^{n_k}x\}_k$ is somewhere dense, but not dense in $X$. That is, $T$ and $\{T^{n_k}\}_k$ do not share the same hypercyclic vectors.