I think we are interpreting the OP differently. In your case, as I read your answer, the executioner never stops, cutting in a circle until they are done. The way I meant for the op to be interpreted is that after the first run of kills, he/she starts back over and kills the new first person, continue through and starting over once they come to the end with the 'next' first person.
It's your puzzle, so your interpretation prevails. However, you mentioned that it's a 'classic' and the prisoners are in a circle, which, in my opinion, strongly suggests the classic interpretation of the problem known as the Josephus problem. The reason they are in a circle is that you don't stop at the end and start the next round with the 'next first person' as if they were in a row. That distinction is what makes this problem interesting, IMHO.