Two motorized 24-hour [light timers](https://en.wikipedia.org/wiki/Time_switch) are [daisy chained](https://en.wikipedia.org/wiki/Daisy_chain_%28electrical_engineering%29) between a power outlet and a light bulb. For these timers, devise schedules and choose initial times that produce the following lighting pattern with the largest possible whole number $n$, beginning when the outlet's power is switched on:

$~~~$ On for 1/2 hour,$~$ off for $n$ hours,
$~~~~$ on for 1/2 hour,$~$ off for $n$ hours,
$~~~~$ on for 1/2 hour,$~$ off for $n$ hours,
$~~~~~~\vdots$

**If you are unfamiliar with these timers**

Each timer repeatedly cycles through its schedule of 24 intervals that last an hour apiece.
•$~$ A circular dial determines the current point in the schedule
•$~$ A motor rotates the dial to advance through its schedule whenever power is supplied to the timer
•$~$ You may initially set the dial to any minute of any interval
•$~$ You may preset each interval to ON or OFF
•$~$ When the dial is in an interval that was set to ON, the timer becomes a direct connection for power to flow to whatever is plugged into the timer
•$~$ When the dial is in an interval that was set to OFF, the timer does not provide a connection between what is plugged into it and what it plugs into
•$~$ The resulting ON and OFF durations can be fractions of an hour if the timer is set to begin within an interval or if incoming power is interrupted during an interval

The first timer is plugged into the outlet.
•$~$ It runs nonstop after the outlet is switched on
•$~$ It supplies power—but only when its dial is in an ON interval—to the second timer

The second timer has the light bulb plugged into it.
•$~$ It advances through its schedule only when the first timer supplies power
•$~$ It lights the bulb, but only while powered by the first timer and when its dial is in an ON interval

Related puzzles
Odd hours with two timers
Day and night of the two timers