Sum of Pairwise Modulo Checksums (i mod j + j mod i) for All Pairs up to n

A distributed network system has `n` data packets whose identifiers are the integers `1` through `n`. For two identifiers `i` and `j`, the checksum of the pair is ```text C(i, j) = (i mod j) + (j mod i) ``` where `a mod b` is the remainder when `a` is divided by `b` (Python's `a % b`). Compute the sum of `C(i, j)` over every pair with `1 <= i <= n` and `1 <= j <= n`, and return it modulo `10^9 + 7`. ### Function Signature ```python def compute_checksum_aggregation(n: int) -> int: ``` ### Rules - Pairs are ordered: `(i, j)` and `(j, i)` are both counted when `i != j`, and the `n` pairs with `i == j` are included as well (each has checksum 0). - Add up all `n * n` checksums exactly, then return the total modulo `1000000007`, an integer in `[0, 1000000006]`. ### Constraints - `1 <= n <= 10^6` - For the largest inputs the unreduced total is far beyond `2^31 - 1` and also beyond `2^53`; only its remainder modulo `10^9 + 7` is returned. ### Examples **Example 1** ```text Input: n = 2 Output: 2 ``` `C(1, 1) = 0`, `C(1, 2) = 1 + 0 = 1`, `C(2, 1) = 0 + 1 = 1` and `C(2, 2) = 0`, for a total of 2. **Example 2** ```text Input: n = 3 Output: 10 ``` | C(i, j) | j = 1 | j = 2 | j = 3 | | --- | --- | --- | --- | | i = 1 | 0 | 1 | 1 | | i = 2 | 1 | 0 | 3 | | i = 3 | 1 | 3 | 0 | For example, `C(2, 3) = (2 mod 3) + (3 mod 2) = 2 + 1 = 3`. The nine values sum to 10. **Example 3** ```text Input: n = 4 Output: 24 ``` Summing `C(i, j)` over `j = 1, 2, 3, 4` gives 3 for `i = 1`, 6 for `i = 2`, 8 for `i = 3` and 7 for `i = 4`, for a total of 24.