Case 1:

The (1,2)-perfect numbers between 1 and 1000 are:
6
28
496

Case 2:

The (2,6)-perfect numbers between 1 and 1000 are:
42
84
160
336

Case 3:

The (2,2)-perfect numbers between 1 and 1000 are:
2
4
16
64

Case 4:

The (1,2)-perfect numbers between 1000 and 10000 are:
8128

Case 5:

The (2,2)-perfect numbers between 1000 and 10000 are:
4096

Case 6:

The (2,6)-perfect numbers between 1000 and 10000 are:
1344

Case 7:

The (1,3)-perfect numbers between 1 and 10000 are:
120
672

Case 8:

The (2,3)-perfect numbers between 1 and 10000 are:
8
21
512

Case 9:

The (3,3)-perfect numbers between 1 and 10000 are:

Case 10:

The (2,2)-perfect numbers between 4096 and 4096 are:
4096