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Count numbers up to N that cannot be expressed as sum of at least two consecutive positive integers
  • Last Updated : 04 May, 2021

Given a positive integer N, the task is to find the count of integers from the range [1, N] such that the integer cannot be expressed as sum of two or more consecutive positive integers.

Examples:

Input: N = 10
Output: 4
Explanation: The integers that cannot be expressed as sum of two or more consecutive integers are {1, 2, 4, 8}. Therefore, the count of integers is 4.

Input: N = 100
Output: 7

Naive Approach: The given problem can be solved based on the observation that if a number is a power of two, then it cannot be expressed as a sum of consecutive numbers. Follow the steps below to solve the given problem:



  • Initialize a variable, say count that stores the count of numbers over the range [1, N] that cannot be expressed as a sum of two or more consecutive integers.
  • Iterate over the range [1, N], and if the number i is a perfect power of 2, then increment the value of count by 1.
  • After completing the above steps, print the value of count as the result.

Below is the implementation of the above approach:

C++




// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to check if a number can
// be expressed as a power of 2
bool isPowerof2(unsigned int n)
{
    // f N is power of
    // two
    return ((n & (n - 1)) && n);
}
 
// Function to count numbers that
// cannot be expressed as sum of
// two or more consecutive +ve integers
void countNum(int N)
{
    // Stores the resultant
    // count of integers
    int count = 0;
 
    // Iterate over the range [1, N]
    for (int i = 1; i <= N; i++) {
 
        // Check if i is power of 2
        bool flag = isPowerof2(i);
 
        // Increment the count if i
        // is not power of 2
        if (!flag) {
            count++;
        }
    }
 
    // Print the value of count
    cout << count << "\n";
}
 
// Driver Code
int main()
{
    int N = 100;
    countNum(N);
 
    return 0;
}
Output: 
7

 

Time Complexity: O(N)
Auxiliary Space: O(1)

Efficient Approach: The above approach can also be optimized based on the observation that the integers which are not powers of 2, except for 2, can be expressed as the sum of two or more consecutive positive integers. Therefore, the count of such integers over the range [1, N] is given by (log2 N + 1).

Below is the implementation of the above approach:

C++




// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to count numbers that
// cannot be expressed as sum of
// two or more consecutive +ve integers
void countNum(int N)
{
    // Stores the count
// of such numbers
    int ans = log2(N) + 1;
 
    cout << ans << "\n";
}
 
// Driver Code
int main()
{
    int N = 100;
    countNum(N);
 
    return 0;
}

Java




// java program for the above approach
import java.io.*;
import java.lang.*;
import java.util.*;
 
public class GFG {
 
    // Function to count numbers that
    // cannot be expressed as sum of
    // two or more consecutive +ve integers
    static void countNum(int N)
    {
       
        // Stores the count
        // of such numbers
        int ans = (int)(Math.log(N) / Math.log(2)) + 1;
 
        System.out.println(ans);
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        int N = 100;
        countNum(N);
    }
}
 
// This code is contributed by Kingash.
Output: 
7

 

Time Complexity: O(log N)
Auxiliary Space: O(1)

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