Count numbers up to N that cannot be expressed as sum of at least two consecutive positive integers
Last Updated :
21 Nov, 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;
}
// Java program for the above approach
import java.util.*;
class GFG{
// Function to check if a number can
// be expressed as a power of 2
static boolean isPowerof2(int n)
{
// f N is power of
// two
return ((n & (n - 1)) > 0 && n > 0);
}
// Function to count numbers that
// cannot be expressed as sum of
// two or more consecutive +ve integers
static 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
boolean flag = isPowerof2(i);
// Increment the count if i
// is not power of 2
if (!flag)
{
count++;
}
}
// Print the value of count
System.out.print(count + "\n");
}
// Driver Code
public static void main(String[] args)
{
int N = 100;
countNum(N);
}
}
// This code is contributed by shikhasingrajput
# Python3 program for the above approach
# Function to check if a number can
# be expressed as a power of 2
def isPowerof2(n):
# if N is power of
# two
return ((n & (n - 1)) and n)
# Function to count numbers that
# cannot be expressed as sum of
# two or more consecutive +ve integers
def countNum(N):
# Stores the resultant
# count of integers
count = 0
# Iterate over the range [1, N]
for i in range(1, N + 1):
# Check if i is power of 2
flag = isPowerof2(i)
# Increment the count if i
# is not power of 2
if (not flag):
count += 1
# Print the value of count
print(count)
# Driver Code
if __name__ == '__main__':
N = 100
countNum(N)
# This code is contributed by mohit kumar 29.
// C# program for the above approach
using System;
class GFG{
// Function to check if a number can
// be expressed as a power of 2
static bool isPowerof2(int n)
{
// f N is power of
// two
return ((n & (n - 1)) > 0 && n > 0);
}
// Function to count numbers that
// cannot be expressed as sum of
// two or more consecutive +ve integers
static 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
Console.Write(count + "\n");
}
// Driver Code
public static void Main(String[] args)
{
int N = 100;
countNum(N);
}
}
// This code is contributed by 29AjayKumar
<script>
// JavaScript program for the above approach
// Function to check if a number can
// be expressed as a power of 2
function isPowerof2(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
function countNum(N)
{
// Stores the resultant
// count of integers
let count = 0;
// Iterate over the range [1, N]
for (let i = 1; i <= N; i++) {
// Check if i is power of 2
let flag = isPowerof2(i);
// Increment the count if i
// is not power of 2
if (!flag) {
count++;
}
}
// Print the value of count
document.write(count + "\n");
}
// Driver code
let N = 100;
countNum(N);
// This code is contributed by souravghosh0416.
</script>
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 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.
# Python 3 program for the above approach
import math
# Function to count numbers that
# cannot be expressed as sum of
# two or more consecutive +ve integers
def countNum(N):
# Stores the count
# of such numbers
ans = int(math.log2(N)) + 1
print(ans)
# Driver Code
if __name__ == "__main__":
N = 100
countNum(N)
# This code is contributed by ukasp.
// C# program for the above approach
using System;
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;
Console.WriteLine(ans);
}
// Driver Code
static void Main()
{
int N = 100;
countNum(N);
}
}
// This code is contributed by SoumikMondal.
<script>
// Javascript program for the above approach
// Function to count numbers that
// cannot be expressed as sum of
// two or more consecutive +ve integers
function countNum(N)
{
// Stores the count
// of such numbers
var ans = parseInt(Math.log(N) / Math.log(2)) + 1;
document.write(ans);
}
// Driver Code
var N = 100;
countNum(N);
// This code contributed by shikhasingrajput
</script>
Time Complexity: O(log N)
Auxiliary Space: O(1)
Similar Reads
DSA Tutorial - Learn Data Structures and Algorithms DSA (Data Structures and Algorithms) is the study of organizing data efficiently using data structures like arrays, stacks, and trees, paired with step-by-step procedures (or algorithms) to solve problems effectively. Data structures manage how data is stored and accessed, while algorithms focus on
7 min read
Quick Sort QuickSort is a sorting algorithm based on the Divide and Conquer that picks an element as a pivot and partitions the given array around the picked pivot by placing the pivot in its correct position in the sorted array. It works on the principle of divide and conquer, breaking down the problem into s
12 min read
Merge Sort - Data Structure and Algorithms Tutorials Merge sort is a popular sorting algorithm known for its efficiency and stability. It follows the divide-and-conquer approach. It works by recursively dividing the input array into two halves, recursively sorting the two halves and finally merging them back together to obtain the sorted array. Merge
14 min read
SQL Commands | DDL, DQL, DML, DCL and TCL Commands SQL commands are crucial for managing databases effectively. These commands are divided into categories such as Data Definition Language (DDL), Data Manipulation Language (DML), Data Control Language (DCL), Data Query Language (DQL), and Transaction Control Language (TCL). In this article, we will e
7 min read
Data Structures Tutorial Data structures are the fundamental building blocks of computer programming. They define how data is organized, stored, and manipulated within a program. Understanding data structures is very important for developing efficient and effective algorithms. What is Data Structure?A data structure is a st
2 min read
Bubble Sort Algorithm Bubble Sort is the simplest sorting algorithm that works by repeatedly swapping the adjacent elements if they are in the wrong order. This algorithm is not suitable for large data sets as its average and worst-case time complexity are quite high.We sort the array using multiple passes. After the fir
8 min read
Breadth First Search or BFS for a Graph Given a undirected graph represented by an adjacency list adj, where each adj[i] represents the list of vertices connected to vertex i. Perform a Breadth First Search (BFS) traversal starting from vertex 0, visiting vertices from left to right according to the adjacency list, and return a list conta
15+ min read
Binary Search Algorithm - Iterative and Recursive Implementation Binary Search Algorithm is a searching algorithm used in a sorted array by repeatedly dividing the search interval in half. The idea of binary search is to use the information that the array is sorted and reduce the time complexity to O(log N). Binary Search AlgorithmConditions to apply Binary Searc
15 min read
Insertion Sort Algorithm Insertion sort is a simple sorting algorithm that works by iteratively inserting each element of an unsorted list into its correct position in a sorted portion of the list. It is like sorting playing cards in your hands. You split the cards into two groups: the sorted cards and the unsorted cards. T
9 min read
Array Data Structure Guide In this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
4 min read