So from the Wikipedia page about this problem, there is a variant of this problem where for $n$ people in a circle, instead of killing the next person, kills the $k$th person instead. (Thus the original problem will have $k=2$.)
Apparently there are solutions for the original $k=2$ which is $f(n)=2(n-2^{\lfloor\log_2n\rfloor})+1$ and for $k=3$ which is $f(n)=3(n-round(\alpha\cdot(3/2)^m))+2$ with $m$ being the biggest number such that $round(\alpha\cdot(3/2)^m)\le n$ and this constant $\alpha\approx0.8111$.
So I was wondering how is the solution for $k=3$ case found (lack of citation) and is there any way to search for solutions for higher $k$? Any research/paper links will be appreciated. Thanks!
For the k=3 case, the paper is:
http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=0EF1E16177A3F71FB825A92AE8245B5A?doi=10.1.1.46.1946&rep=rep1&type=pdf
general problem:
https://www.jstor.org/stable/43686318?casa_token=flGruEZzecEAAAAA%3A-H4adfhYl4mJ7kTdv61n0VclU2xdXE_OcBOk7qK2ewmko-HbXSdCMfJlZpQVGbNZwglHDQo-iEzoSwfdIPcnfZNmG9FfAyNWaZQsK-BrZoU0t6QwZg&seq=1#metadata_info_tab_contents
Another variant where each person has $l$ lives:
https://link.springer.com/article/10.1007/s00224-011-9343-6