That argument sounds correct to me, at least for the non-degenerate cases where n >= 2. For n=1, the lights start out all-on, which solves it provided you're allowed to make zero moves. Arguably there are no legal moves when n=1 since there is no valid d. Question: Given an initial configuration of all-off (or all-on), are all of the zero COL patterns attainable? I think http://oeis.org/A103314 gives the count if you include the all-off case, which http://oeis.org/A070894 does not. Tom Don Reble writes:
"Given a circle of n lights, exactly one of which is initially on, it is permitted to change the state of a bulb provided that one also changes the state of every dth bulb after it (where d is a divisor of n strictly less than n), provided that all n/d bulbs were originally in the same state as one another. Is it possible to turn all the bulbs on by making a sequence of moves of this kind?"
OEIS http://oeis.org/A103314 looks very apropos.
A070894 looks even better.
The set of bulbs changed by a move has a center-of-light at zero. (Arrange the bulbs evenly around the circle. And then all go on or all go off.) The original center is non-zero, and a move changes it to a weighted combination of zero and the old value: but never to zero.
So you can't even turn them all off. Please refute this, if you're the last one to bed.
-- Don Reble djr@nk.ca