Eliminating ambiguity when referring to complexity in cases of multiple parameters and/or multiple results
You can report:


The running time (e.g., $O(k)$ vs $O(n)$),
The space requirements (e.g., $O(1)$, $O(k)$, etc.), and
Whether the algorithm is a streaming (online) algorithm or a batch algorithm.   A batch algorithm returns the entire answer at once.  A streaming or online algorithm returns numbers one at a time.  You might also be interested in the notion of "delay"; for instance, a $O(\log n)$ delay algorithm returns algorithms one at a time, taking at most $O(\log n)$ steps (running time) to output each number.


Since there are two variables here, $n$ and $k$, if you report a running time like $O(k)$, it may be helpful to mention explicitly that the constant is independent of $n$.

Formal papers aren't always as formal as you might think.  They'll use standard notation and terminology where they exist, but if standard notions don't cover it, they'll just use the English language to describe the situation, or they may introduce some new terminology and provide a definition for it.  That's the great thing about human language: it provides a way to express arbitrary new concepts, even ones we don't already have a pre-existing standard name for.