1. j is used in some circumstances to represent so it's not the ideal pronumeral to use here.
2. j could be confused with J which is a common symbol for rational numbers. So again, not the ideal choice of pronumeral.
3. It needs to be made very clear that the j represents the set of integers.
4. The values of these integers is not restricted to 0, 1, ... n. In fact, the required values can be any n consecutive integers, including negative values.