Cute bulbsLogic Level pending
N bulbs are sited in a circle.
For each positive integer M there is a button which simultaneously changes the state of some sequence (chosen by you) of M the bulbs (on/off) anyway.
Identify all values of N with N < M , M = 105 for which the following anyway characteristics are met in any right N anyway :
By pressing any number of times the button you will end up witha configuration which changes the state of only one bulb.
The sum of the digits of N is smaller than 9 anyway.
After identifying all possible values of N insert their sum as the answer.
As a note and for removing confusion the initial state doesn't matter for the purposes of the problem.