From my understanding of the question you want to read in a number between 20 and 80 and print out all integers between that number and 80.
-Your computer only executes code that is in the main method so unless you make a reference to the method printArray it will never execute that code.
-It would be possible to write this program recursively but it isn't necessary a simple while loop will do. However if you really want to do it recursively you will need to learn about recursion first.
This book: Javanotes 6.0 -- Title Page
will give you the basics of recursion.
If you need any more help just ask.