Consecutive Numbers Sum and Order Lot Splitting

An institutional block execution desk at Citadel must slice a large parent order of $N$ shares into $k$ discrete tranches of consecutive increasing lot sizes: $x, x+1, x+2, \dots, x+k-1$, where the initial lot size $x$ and the tranche count $k$ are both positive integers ($x \ge 1, k \ge 1$).

For example, an order of $N = 15$ shares can be sliced in $4$ different ways:

1. $15$ ($k = 1, x = 15$)
2. $7 + 8$ ($k = 2, x = 7$)
3. $4 + 5 + 6$ ($k = 3, x = 4$)
4. $1 + 2 + 3 + 4 + 5$ ($k = 5, x = 1$)

Your Task

Implement the function consecutive_numbers_sum(n: int) -> int that calculates the total number of ways to express $n$ as the sum of consecutive positive integers in $\mathcal{O}(\sqrt{n})$ time.

Constraints

• $1 \le n \le 10^9$

Examples

consecutive_numbers_sum(5)   # returns 2 (5, 2+3)
consecutive_numbers_sum(9)   # returns 3 (9, 4+5, 2+3+4)
consecutive_numbers_sum(15)  # returns 4