Multiples of hypercyclic operators

We give a negative answer to a question of Prăjitură by showing that there exists an invertible bilateral weighted shift $T$ on $\ell_2(\mathbb{Z})$ such that $T$ and $3T$ are hypercyclic but $2T$ is not.

Moreover, any $G_\delta$ set $M \subseteq (0,\infty)$ which is bounded and bounded away from zero can be realized as $M=\{t>0 \mid tT \textrm{ is hypercyclic}\}$ for some invertible operator $T$ acting on a Hilbert space.