  A cohesive power of a computable structure is an effective ultrapower where a
cohesive set acts as an ultrafilter. Let $\omega$, $\zeta$, and $\eta$ denote
the respective order-types of the natural numbers, the integers, and the
rationals. We study cohesive powers of computable copies of $\omega$ over
$\Delta_2$ cohesive sets. We show that there is a computable copy $\mathcal{L}$
of $\omega$ such that, for every $\Delta_2$ cohesive set $C$, the cohesive
power of $\mathcal{L}$ over $C$ has order-type $\omega + \eta$. This improves
an earlier result of Dimitrov, Harizanov, Morozov, Shafer, A. Soskova, and
Vatev by generalizing from $\Sigma_1$ cohesive sets to $\Delta_2$ cohesive sets
and by computing a single copy of $\omega$ that has the desired cohesive power
over all $\Delta_2$ cohesive sets. Furthermore, our result is optimal in the
sense that $\Delta_2$ cannot be replaced by $\Pi_2$. More generally, we show
that if $X \subseteq \mathbb{N} \setminus \{0\}$ is a Boolean combination of
$\Sigma_2$ sets, thought of as a set of finite order-types, then there is a
computable copy $\mathcal{L}$ of $\omega$ where the cohesive power of
$\mathcal{L}$ over any $\Delta_2$ cohesive set has order-type $\omega +
\sigma(X \cup \{\omega + \zeta\eta + \omega^*\})$. If $X$ is finite and
non-empty, then there is also a computable copy $\mathcal{L}$ of $\omega$ where
the cohesive power of $\mathcal{L}$ over any $\Delta_2$ cohesive set has
order-type $\omega + \sigma(X)$. An unexpected byproduct of our work is a new
method for constructing infinite $\Pi_1$ sets that do not have $\Delta_2$
cohesive subsets. In fact, we construct an infinite $\Pi_1$ set that does not
have a $\Delta_2$ p-cohesive subset. Infinite $\Pi_1$ sets without $\Delta_2$
r-cohesive subsets generalize D. Martin's classic co-infinite c.e. set with no
maximal superset and have appeared in the work of Lerman, Shore, and Soare.
