  Let $S \subseteq \mathbb{N}$ have the property that for each $k \in S$ the
set $(S - k) \cap \mathbb{N} \setminus S$ has asymptotic density $0$. We prove
that there exists a basic sequence $Q$ where the set of numbers $Q$-normal of
all orders in $S$ but not $Q$-normal of all orders not in $S$ has full
Hausdorff dimension. If the function $k \mapsto 1_S(k)$ is computable, then
there exist computable examples. For example, there exists a computable basic
sequence $Q$ where the set of numbers normal of all even orders and not normal
of all odd orders has full Hausdorff dimension. This is in strong constrast to
the $b$-ary expansions where any real number that is normal of order $k$ must
also be normal of all orders between $1$ and $k-1$.
  Additionally, all numbers we construct satisfy the unusual condition that
block frequencies sampled along non-trivial arithmetic progressions don't
converge to the expected value. This is also in strong contrast to the case of
the $b$-ary expansions, but more similar to the case of the continued fraction
expansion. As a corollary, the set of $Q$-normal numbers that are not normal
when sampled along any non-trivial arithmetic progression has full Hausdorff
dimension.
