Has the Josephus sequence $J(n,1),J(n,2),\dots$ been studied from a coverage viewpoint?

I have been investigating an empirical variant of the classical Josephus problem and would like to know whether it has been studied previously.

Let $J(n,k)$ denote the survivor of the classical Josephus problem with population size $n$ and elimination interval $k$.

For fixed $n$, instead of fixing $k$, I consider the sequence

$$ J(n,1),J(n,2),J(n,3),\ldots $$

and define the coverage function $C_n(k)$, which is the number of distinct survivor positions discovered after considering elimination intervals from $1$ to $k$.

$$ C_n(k)= |\{J(n,1),J(n,2),\dots,J(n,k)\}| $$

I then define

$$ M(n)=\min\{k : C_n(k)=n\}, $$

the smallest elimination interval by which every possible survivor position has been observed.

Empirically, I have computed $M(n)$ for all $1\le n\le11000$. Every value tested achieves complete coverage, but the manner in which coverage is achieved varies considerably. Most values of $n$ exhibit fairly steady growth of $C_n(k)$, whereas others exhibit long plateaus during which increasing $k$ produces no new survivor positions before another survivor eventually appears.

My question is simply:

Has this coverage (or first-occurrence) viewpoint of the Josephus survivor function been studied before?

More specifically, are there any known results concerning:

• the sequence $J(n,1),J(n,2),\ldots$;
• the growth of $M(n)$;
• or the occurrence of long plateaus in the coverage function $C_n(k)$?

I’d be very grateful for any references, even if this work appears under different terminology.