Problem of the Week [July 8, 2013]

A school has a row of  lockers, numbered 1 through . After arriving at school one day, Josephine starts at the beginning of the row and closes every second locker until she reaches the end of the row. Then on her way back, she closes every second locker that is still open. She continues in this manner along the row, until only one locker remains open. Define  to be the last open locker. Prove that there are infinitely many positive integers  such that .

Correct answers as well as solutions will be acknowledged.