Check if a number can be expressed as a sum of consecutive numbers - GeeksforGeeks

21 Jan, 2022
Given a number n, the task is to check whether it can be expressed as a sum of two or more consecutive numbers or not. Examples:  
Input  : n = 10 
Output : true
It can be expressed as sum of two consecutive
numbers 1 + 2 + 3 + 4.

Input  : n = 16  
Output : false
It cannot be expressed as sum of two consecutive
numbers.

Input  : n = 5  
Output : true
2 + 3 = 5
 
There is a direct and quick method to solve this. If a number is a power of two, then it cannot be expressed as a sum of consecutive numbers otherwise Yes.The idea is based on below two facts. 1) Sum of any two consecutive numbers is odd as one of them has to be even and the other odd. 2) 2n = 2n-1 + 2n-1If we take a closer look at 1) and 2), we can get the intuition behind the fact.Below is the implementation of the above idea.  
C++
Java
Python3
C#
PHP
Javascript
// C++ program to check if a number can// be expressed as sum of consecutive numbers#include<bits/stdc++.h>using namespace std;  // This function returns true if n can be// expressed sum of consecutive.bool canBeSumofConsec(unsigned int n){    // We basically return true if n is a    // power of two    return ((n&(n-1)) && n);}  // Driver codeint main(){    unsigned int n = 15;    canBeSumofConsec(n)? cout << "true" :                         cout << "false";    return 0;}
// Java program to check if a number can// be expressed as sum of consecutive numbers  class Test{    // This function returns true if n can be    // expressed sum of consecutive.    static boolean canBeSumofConsec(int n)    {        // We basically return true if n is a        // power of two        return (((n&(n-1))!=0) && n!=0);    }          // Driver method    public static void main(String[] args)     {        int n = 15;        System.out.println(canBeSumofConsec(n) ? "true" : "false");    }}
# Python 3 program to check if a number can# be expressed as sum of consecutive numbers    # This function returns true if n # can be expressed sum of consecutive.def canBeSumofConsec(n) :      # We basically return true if n is a    # power of two    return ((n&(n-1)) and n)    # Driver coden = 15if(canBeSumofConsec(n)) :    print("true")else :    print("false")      # This code is contributed by Nikita Tiwari.
// C# program to check if a number can be// expressed as sum of consecutive numbersusing System;  class Test{    // This function returns true if n    // can be expressed sum of consecutive.    static bool canBeSumofConsec(int n)    {        // We basically return true if n is a        // power of two        return (((n & (n - 1)) != 0) && n != 0);    }          // Driver Code    public static void Main()     {        int n = 15;        Console.Write(canBeSumofConsec(n) ? "True" : "False");    }}  // This code is contributed by Nitin Mittal.
<?php// php program to check if a number// can be expressed as sum of // consecutive numbers  // This function returns true if n// can be expressed sum of consecutive.function canBeSumofConsec($n){          // We basically return true if n is a    // power of two    return (($n & ($n - 1)) && $n);}  // Driver code    $n = 15;    if(canBeSumofConsec($n))         echo "true" ;    else        echo "false";  // This code is contributed by// nitin mittal. ?>
<script>  // Javascript program to check if a number can// be expressed as sum of consecutive numbers        // This function returns true if n can be    // expressed sum of consecutive.    function canBeSumofConsec(n)    {        // We basically return true if n is a        // power of two        return (((n&(n-1))!=0) && n!=0);    }                // function call          let n = 15;    document.write(canBeSumofConsec(n) ? "true" : "false");      </script>
Output: 
True
Time Complexity: O(1)
Auxiliary Space: O(1)
Let number chosen to represent N as a sum of consecutive numbers be X + 1, X + 2, X + 3 .... Y 
Sum of these chosen numbers = Sum of first Y natural numbers – Sum of first X natural number 
Sum of first Y natural number = Sum of first X natural number = We know that,N = Sum of first Y natural number – Sum of first X natural numberLet Y – X = a, Y + X + 1 = bY + X + 1 > Y – X, b > a, 2N = a * bIt means that a and b are factor of 2N, we know that X and Y are integers so,1. b – a – 1 => multiple of 2 (Even number)2. b + a + 1 => multiple of 2 (Even number)
Both conditions must be satisfied
From 1 and 2 we can say that either one of them (a, b) should be Odd and another one Even
So if the number (2N) has only odd factors (can’t be possible as it is an even number (2N not N) ) or only even factors we can’t represent it as a sum of any consecutive natural numbers
So now, we have to now only check whether it has an odd factor or not
1. If the number (2N not N) does not have any odd factor (contains only even factor means can be represented as) then we can’t represent it as a sum of consecutive number
2. If the number (2N not N) has an odd factor then we can represent it as a sum of a consecutive number
After this we have to only check whether we can represent (2N as ) or not
if Yes then answer is false or 0
if No then answer is true or 1
Below is the implementation of the above idea : 
C++14
C
Java
Python3
C#
Javascript
#include <bits/stdc++.h>using namespace std;  long long int canBeSumofConsec(long long int n) {     // Updating n with 2n    n = 2 * n;    // (n & (n - 1)) => Checking whether we can write 2n as 2^k    // if yes (can't represent 2n as 2^k) then answer 1    // if no (can represent 2n as 2^k) then answer 0    return ((n & (n - 1)) != 0);}  int main(){    long long int n = 10;    cout<<canBeSumofConsec(n)<<"\n";}
#include <stdio.h>  long long int canBeSumofConsec(long long int n) {     // Updating n with 2n    n = 2 * n;    // (n & (n - 1)) => Checking whether we can write 2n as 2^k    // if yes (can't represent 2n as 2^k) then answer 1    // if no (can represent 2n as 2^k) then answer 0    return ((n & (n - 1)) != 0);}  int main(){    long long int n = 10;    printf("%lld", canBeSumofConsec(n));}
import java.util.*;class GFG{   static int canBeSumofConsec( int n) {     // Updating n with 2n    n = 2 * n;         // (n & (n - 1)) => Checking whether we can write 2n as 2^k    // if yes (can't represent 2n as 2^k) then answer 1    // if no (can represent 2n as 2^k) then answer 0    return ((n & (n - 1)) != 0)?1:0;}  public static void main(String[] args){    int n = 10;    System.out.print(canBeSumofConsec(n)+"\n");}}  // This code is contributed by umadevi9616 
def canBeSumofConsec(n):        # Updating n with 2n    n = 2 * n;      # (n & (n - 1)) => Checking whether we can write 2n as 2^k    # if yes (can't represent 2n as 2^k) then answer 1    # if no (can represent 2n as 2^k) then answer 0    if((n & (n - 1)) != 0):        return 1;    else:        return 0;  if __name__ == '__main__':    n = 10;    print(canBeSumofConsec(n));  # This code is contributed by umadevi9616 
using System;  public class GFG {      static int canBeSumofConsec(int n)    {                // Updating n with 2n        n = 2 * n;          // (n & (n - 1)) => Checking whether we can write 2n as 2^k        // if yes (can't represent 2n as 2^k) then answer 1        // if no (can represent 2n as 2^k) then answer 0        return ((n & (n - 1)) != 0) ? 1 : 0;    }      public static void Main(String[] args) {        int n = 10;        Console.Write(canBeSumofConsec(n) + "\n");    }}  // This code is contributed by umadevi9616
<script>      function canBeSumofConsec(n) {        // Updating n with 2n        n = 2 * n;          // (n & (n - 1)) => Checking whether we can write 2n as 2^k        // if yes (can't represent 2n as 2^k) then answer 1        // if no (can represent 2n as 2^k) then answer 0        return ((n & (n - 1)) != 0) ? 1 : 0;    }          var n = 10;        document.write(canBeSumofConsec(n) + "\n");  // This code is contributed by umadevi9616</script>
1
Time Complexity: O(1)
Auxiliary Space: O(1)
Reference: http://www.cut-the-knot.org/arithmetic/UnpropertyOfPowersOf2.shtmlThis article is contributed by Sahil Chhabra. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. 
nitin mittal
susmitakundugoaldanga
anshkush92
umadevi9616
subhammahato348
Bit Magic
Mathematical
Mathematical
Bit Magic
Writing code in comment?
Please use ide.geeksforgeeks.org,
generate link and share the link here.
Comments
Old Comments
Cyclic Redundancy Check and Modulo-2 Division
Little and Big Endian Mystery
Binary representation of a given number
Bit Fields in C
Bits manipulation (Important tactics)
Program for Fibonacci numbers
C++ Data Types
Write a program to print all permutations of a given string
Set in C++ Standard Template Library (STL)
Coin Change | DP-7