829. Consecutive Numbers Sum

829. Consecutive Numbers Sum

Hard

Given a positive integer N, how many ways can we write it as a sum of consecutive positive integers?

Example 1:

Input: 5
Output: 2
Explanation: 5 = 5 = 2 + 3

Example 2:

Input: 9
Output: 3
Explanation: 9 = 9 = 4 + 5 = 2 + 3 + 4

Example 3:

Input: 15
Output: 4
Explanation: 15 = 15 = 8 + 7 = 4 + 5 + 6 = 1 + 2 + 3 + 4 + 5

Note: 1 <= N <= 10 ^ 9.

---------------------
求等差数列为1，和为N的数列的个数，假设数列的首项为x，项数是m，则如果存在这一数列，N = (x + (x + m - 1)) * m / 2. 那么我们就遍历m的可能性

O(lgN)
ref: https://zhanghuimeng.github.io/post/leetcode-829-consecutive-numbers-sum/

class Solution {
    public int consecutiveNumbersSum(int N) {
        int rt = 0;
        
        for (int m = 1; ; m++) {
            int mx = N - (m - 1) * m / 2;
            if (mx <= 0) break;
            if (mx % m == 0) rt++;
        }
        
        return rt;
    }
}