For integer \(n\geq2\), I have \(n\) lightbulbs arranged on the circumference of a circle, each lightbulb can be either on or off. When I press a button, the lightbulbs will either:
###### Image Credit: Flickr patrick etter.

- once every minute, be turned on if it is in the same state as the one of its left (both off or both on),
- once every minute, be turned off if it is in the different state as the one of its left (one on and one off).

How many integers \(n < 10^6 \) are there such that after at most \(n\) minutes, all the lightbulbs will be on simultaneously regardless of their initial states?

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