Tasker is a useful automation tool for Android that allows programmable events to occur.

By default, Tasker will use the device clock/alarm function to guarantee events are run. This function is called Reliable Alarms. A clock will be visible on the Android notification bar and other screens, even though no actual alarm is set. This feature can be disabled by accessing:

Preferences > Monitor > Use Reliable Alarms

…and set it to Never, or only when the device is off.