Consecutive Numbers Sum

Count how many ways a given n (1 ≤ n ≤ 10^9) can be written as a sum of consecutive positive integers. The core challenge is finding lengths k and starting values a satisfying n = k*(2a+k-1)/2 — which reduces to a number-theoretic/divisor-counting check (equivalently counting appropriate divisors of n) that can be done in O(sqrt(n)).