163200

First I tried to find out which lights turned on when which numbers were put in, and I thought that I should give each light a number.
It looked a bit like this:
1: 1, 2, 4
2: 3
3: 2
4: 3, 4
5: none
and so on. I then listed each light with the numbers that light it up. This is what I got:
light 1: 1, 6, 11, 16...
light 2: 1, 3, 6, 10, 15...
light 3: 1, 4, 9, 16...
light 4: 1, 4, 9...
It is now quite clear to see that the first light lights up when you put in a number 1 more than a multiple of five. The second one is for triangular numbers, the third for even numbers, and the fourth for square numbers.