  An increasing sequence of positive integers $(n_k)_{k\ge 0}$ is said to be a
Jamison sequence if the following property holds true: for every separable
complex Banach space $X$ and every $T\in \mathcal{B}(X)$ which is partially
power-bounded with respect to $(n_k)_{k\ge 0}$, the set $\sigma_p(T)\cap \T$ is
at most countable. We prove that a separable infinite-dimensional complex
Banach space $X$ which admits an unconditional Schauder decomposition is such
that for any sequence $(n_k)_{k\ge 0}$ which is not a Jamison sequence, there
exists $T\in \mathcal{B}(X)$ which is partially power-bounded with respect to
this sequence and such that the set $\sigma_p(T)\cap \T$ is uncountable. We
also investigate the notion of Jamison sequences for $C_0$-semigroups and we
give an arithmetic characterization of these sequences.
