Consecutive Numbers Sum - Leetcode Solution

Problem Statement: Given an integer N, you need to find the number of ways to represent it as a sum of two or more consecutive positive integers.

Example:

Input: N = 15

Output: 4

Explanation: 15 can be represented as: 1+2+3+4+5 = 15 4+5+6 = 15 7+8 = 15 15 = 15

Solution:

We can solve this problem efficiently by using the concept of arithmetic progression. An arithmetic progression is a sequence of numbers in which each element differs from the previous one by a constant value called the common difference.

Suppose we want to represent N as the sum of a consecutive sequence of k numbers, starting from x. Then the sum of this sequence can be represented as:

sum = kx + k(k-1)/2

We need to find the values of x and k such that this equation is satisfied and the sum is equal to N. Rearranging the equation, we get:

x = (N - k*(k-1)/2)/k

We can iterate over all possible values of k and calculate the corresponding value of x. If x is an integer, we have found a valid sequence.

Here is the code to implement this solution:

class Solution:
    def consecutiveNumbersSum(self, N: int) -> int:
        count = 0
        for k in range(1, N+1):
            # calculate x using the formula
            x = (N - k*(k-1)//2) / k
            # check if x is an integer and greater than 0
            if x.is_integer() and x > 0:
                count += 1
        return count

Time Complexity: O(N) as we are iterating over all possible values of k.

Space Complexity: O(1) as we are not using any extra space.

This solution gives us the number of ways to represent N as the sum of two or more consecutive positive integers.

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