I do not know why the program is wrong.it can not end.

Quote:

I do not know the reason.

Read your code carefully!

#include <stdio.h>

int main() {
    int N;
    int arr[20] = {};
    scanf("%d", &N);
    for (int i = 1; i <= N; i++)  // i is the main loop
    {
        int sum = 0;
        int n = 0;
        for (int j = 1; j < i; j++)
        {
            if (i % j == 0)
            {
                sum += j;
                *(arr+n) = j;
                n++;
            }
                
        }
        if (i == sum)
        {
            printf("%d its factors are ", i);
            for (int i = 0; i < n; i++) // and i is this inner loop
            {
                printf("%d ", arr[i]);
            }
            printf("\n");
        }
        else
            continue;
    }
    return 0;
}

The i in inner loop kills the main loop every times you get a result.

As Griff correctly pointed out, your problem is in the following lines

*(arr+n) = j;
  n++;

You don't know in advance how much n will grow (albeit it cannot be greater than N*N), so arr overrun is easily achieved.

If you are actually using C++ (as the tag implies) then choose a vector, instead of C-like array. On the other hand, if you are using C then you should use dynamic memory allocation, calling realloc (see, for instance realloc in c example | How realloc works | return value of realloc| realloc code[^]) whenever arr needs to grow.

The strange array access can also be written in a simpler and more readable way:

// *(arr + n) = j;
arr[n] = j;

As OriginalGriff already stated else continue; can be removed completely.

To find the error it makes sense to monitor the array size.
First I would calculate in a variable how many elements the array arr can hold:

int arr[20] = {};
size_t arr_size = sizeof(arr) / sizeof(arr[0]);

Before each access to the field, we can now check whether the index is still in the field or not:

if (n < arr_size) {
  // *(arr + n) = j;
  arr[n] = j;
}
else
  printf("error: n is too big %d\n", n);
}

As you can easily see the field arr e.g. for N=360 with only 20 elements is not enough to store all numbers and the field overflows.

To look at the problem you can of course output all N that are too large:

if(n >= arr_size)
  printf("for i=%d, n was %d\n", i, n);
if (i == sum) {

for i=360, n was 23

For N to 1000, the arr field would have to contain at least 32 elements.