Arduino MEGA2560 code stops after power down and up

Sorry. I will try to explain better.

Say the Arduino was running a mini LED screen and then the Arduino was unplugged from the computer and so the Arduino had no power and the LED screen had no power. And then you plug in the Arduino again, so they both get power again. The Arduino stops printing to the screen. It only starts again if I upload the code to the Arduino again, or if I open the serial monitor on the computer. How do I get the Arduino to keep printing to the screen on power up?