Sum of all the prime numbers with the maximum position of set bit ≤ D

Given an integer D, the task is to find the sum of all the prime numbers whose maximum position of set bits (the farthest set bit from the right) is less than or equal to D. 
Note: 2 in binary is 10 and the maximum set bit position is 2. 7 in binary is 111, maximum set bit position is 3.

Examples:

Input: D = 3 
Output: 17 
2, 3, 5 and 7 are the only primes 
which satisfy the given condition.
Input: D = 8 
Output: 6081  

Brute Force Approach:

The brute force approach involves checking each number up to 2^D – 1 for primality and then checking its binary representation for the maximum set bit position.

Here’s the approach:

1. Initialize a variable ans to 0.
2. Loop through each number i from 2 to 2^D – 1.
3. For each number i, check if it is prime using a primality test such as trial division.
4. If the number i is prime, convert it to binary and find the maximum set bit position using the bit manipulation techniques.
5. If the maximum set bit position is less than or equal to D, add the number i to ans.
6. Return the value of ans.

C++

#include <bits/stdc++.h>
using namespace std;

bool isPrime(int n) {
    if (n <= 1) {
        return false;
    }
    for (int i = 2; i <= sqrt(n); i++) {
        if (n % i == 0) {
            return false;
        }
    }
    return true;
}

int sumPrime(int d) {
    int n = (1 << d) - 1;
    int ans = 0;
    for (int i = 2; i <= n; i++) {
        if (isPrime(i)) {
            int num = i;
            int msb = 0;
            while (num > 0) {
                msb++;
                num >>= 1;
            }
            if (msb <= d) {
                ans += i;
            }
        }
    }
    return ans;
}

int main() {
    int d = 8;
    cout << sumPrime(d) << endl;
    return 0;
}

Java

import java.lang.Math;

public class Main {
    // Define a function to check if a number is prime
    static boolean isPrime(int n) {
        if (n <= 1) {
            return false;
        }
        for (int i = 2; i <= Math.sqrt(n); i++) {
            if (n % i == 0) {
                return false;
            }
        }
        return true;
    }

    // Define a function to find the sum of all primes whose binary representation has at most d digits
    static int sumPrime(int d) {
        int n = (1 << d) - 1;
        int ans = 0;
        for (int i = 2; i <= n; i++) {
            if (isPrime(i)) {
                int num = i;
                int msb = 0;
                while (num > 0) {
                    msb++;
                    num >>= 1;
                }
                if (msb <= d) {
                    ans += i;
                }
            }
        }
        return ans;
    }

    public static void main(String[] args) {
        int d = 8;
        System.out.println(sumPrime(d));
    }
}

Python3

import math

# Function to check if a given number is prime or not
def isPrime(n):
    if n <= 1:
        return False
    for i in range(2, int(math.sqrt(n)) + 1):
        if n % i == 0:
            return False
    return True

# Function to calculate the sum of all prime numbers less than 2^d
def sumPrime(d):
    n = (1 << d) - 1
    ans = 0
    for i in range(2, n + 1):
        if isPrime(i):
            num = i
            msb = 0
            while num > 0:
                msb += 1
                num >>= 1
            if msb <= d:
                ans += i
    return ans

# Example usage of the functions
d = 8
print(sumPrime(d))

C#

using System;

class MainClass {
    // Check if a given number is prime
    static bool IsPrime(int n)
    {
        if (n <= 1) { // 1 is not prime
            return false;
        }
        for (int i = 2; i <= Math.Sqrt(n);
             i++) { // Check divisors up to sqrt(n)
            if (n % i == 0) {
                return false; // n is divisible by i, so
                              // it's not prime
            }
        }
        return true; // n is only divisible by 1 and itself,
                     // so it's prime
    }

    // Compute the sum of all prime numbers with binary
    // representation up to d bits
    static int SumPrime(int d)
    {
        int n = (1 << d)
                - 1; // Maximum number with d binary digits
        int ans = 0; // Initialize sum to zero
        for (int i = 2; i <= n;
             i++) { // Check all numbers from 2 to n
            if (IsPrime(i)) { // If i is prime
                int num = i;
                int msb = 0;
                while (num > 0) { // Count the number of
                                  // binary digits in i
                    msb++;
                    num >>= 1;
                }
                if (msb <= d) { // If i has at most d binary
                                // digits
                    ans += i; // Add i to the sum
                }
            }
        }
        return ans; // Return the final sum
    }

    static void Main()
    {
        int d = 8; // Set the number of binary digits to 8
        Console.WriteLine(SumPrime(
            d)); // Compute and print the sum of primes
    }
}
// This code is contributed by sarojmcy2e

Javascript

// Function to check if a given number is prime or not
function isPrime(n) {
    if (n <= 1) {
        return false;
    }
    for (let i = 2; i <= Math.sqrt(n); i++) {
        if (n % i == 0) {
            return false;
        }
    }
    return true;
}

// Function to calculate the sum of all prime numbers less than 2^d
function sumPrime(d) {
    let n = (1 << d) - 1;
    let ans = 0;
    for (let i = 2; i <= n; i++) {
        if (isPrime(i)) {
            let num = i;
            let msb = 0;
            while (num > 0) {
                msb++;
                num >>= 1;
            }
            if (msb <= d) {
                ans += i;
            }
        }
    }
    return ans;
}

// Example usage of the functions
let d = 8;
console.log(sumPrime(d));

Output

6081

Time Complexity: O(2^D * D * sqrt(N))
Auxiliary Space: O(1)

Approach: The maximum number which satisfies the given condition is 2^D – 1. So, generate all prime numbers using Sieve of Eratosthenes up to 2^D – 1 then find the sum of all the prime numbers in the same range.

Below is the implementation of the above approach:  

C++

// C++
#include <bits/stdc++.h>
using namespace std;

// Function for Sieve of Eratosthenes
void sieve(bool prime[], int n)
{
    prime[0] = false;
    prime[1] = false;
    for (int p = 2; p * p <= n; p++) {
        if (prime[p] == true) {
            for (int i = p * p; i <= n; i += p)
                prime[i] = false;
        }
    }
}

// Function to return the sum of
// the required prime numbers
int sumPrime(int d)
{

    // Maximum number of the required range
    int maxVal = pow(2, d) - 1;

    // Sieve of Eratosthenes
    bool prime[maxVal + 1];
    memset(prime, true, sizeof(prime));
    sieve(prime, maxVal);

    // To store the required sum
    int sum = 0;

    for (int i = 2; i <= maxVal; i++) {

        // If current element is prime
        if (prime[i]) {
            sum += i;
        }
    }

    return sum;
}

// Driver code
int main()
{
    int d = 8;

    cout << sumPrime(d);

    return 0;
}

Java

// Java implementation of the approach
import java.util.*;

class GFG
{

// Function for Sieve of Eratosthenes
static void sieve(boolean prime[], int n)
{
    prime[0] = false;
    prime[1] = false;
    for (int p = 2; p * p <= n; p++)
    {
        if (prime[p] == true)
        {
            for (int i = p * p;
                     i <= n; i += p)
                prime[i] = false;
        }
    }
}

// Function to return the sum of
// the required prime numbers
static int sumPrime(int d)
{

    // Maximum number of the required range
    int maxVal = (int) (Math.pow(2, d) - 1);

    // Sieve of Eratosthenes
    boolean []prime = new boolean[maxVal + 1];
    Arrays.fill(prime, true);
    sieve(prime, maxVal);

    // To store the required sum
    int sum = 0;

    for (int i = 2; i <= maxVal; i++)
    {

        // If current element is prime
        if (prime[i])
        {
            sum += i;
        }
    }
    return sum;
}

// Driver code
public static void main(String[] args)
{
    int d = 8;

    System.out.println(sumPrime(d));
}
}

// This code is contributed by PrinciRaj1992

Python 3

# Python 3 implementation of the approach
from math import sqrt, pow

# Function for Sieve of Eratosthenes
def sieve(prime, n):
    prime[0] = False
    prime[1] = False
    for p in range(2, int(sqrt(n)) + 1, 1):
        if (prime[p] == True):
            for i in range(p * p, n + 1, p):
                prime[i] = False

# Function to return the sum of
# the required prime numbers
def sumPrime(d):

    # Maximum number of the required range
    maxVal = int(pow(2, d)) - 1;

    # Sieve of Eratosthenes
    prime = [True for i in range(maxVal + 1)]

    sieve(prime, maxVal)

    # To store the required sum
    sum = 0

    for i in range(2, maxVal + 1, 1):

        # If current element is prime
        if (prime[i]):
            sum += i

    return sum

# Driver code
if __name__ == '__main__':
    d = 8

    print(sumPrime(d))

# This code is contributed by Surendra_Gangwar

C#

// C# implementation of the approach
using System;
using System.Linq;

class GFG
{

// Function for Sieve of Eratosthenes
static void sieve(Boolean []prime, int n)
{
    prime[0] = false;
    prime[1] = false;
    for (int p = 2; p * p <= n; p++)
    {
        if (prime[p] == true)
        {
            for (int i = p * p;
                    i <= n; i += p)
                prime[i] = false;
        }
    }
}

// Function to return the sum of
// the required prime numbers
static int sumPrime(int d)
{

    // Maximum number of the required range
    int maxVal = (int) (Math.Pow(2, d) - 1);

    // Sieve of Eratosthenes
    Boolean []prime = new Boolean[maxVal + 1];

    for (int i = 0; i <= maxVal; i++)
        prime.SetValue(true,i);
    sieve(prime, maxVal);

    // To store the required sum
    int sum = 0;

    for (int i = 2; i <= maxVal; i++)
    {

        // If current element is prime
        if (prime[i])
        {
            sum += i;
        }
    }
    return sum;
}

// Driver code
public static void Main(String[] args)
{
    int d = 8;

    Console.WriteLine(sumPrime(d));
}
}

// This code is contributed by 29AjayKumar

Javascript

<script>
//Javascript implementation of the approach

// Function for Sieve of Eratosthenes
function sieve(prime, n)
{
    prime[0] = false;
    prime[1] = false;
    for (var p = 2; p * p <= n; p++) {
        if (prime[p] == true) {
            for (var i = p * p; i <= n; i += p)
                prime[i] = false;
        }
    }
}

// Function to return the sum of
// the required prime numbers
function sumPrime(d)
{

    // Maximum number of the required range
    var maxVal = Math.pow(2, d) - 1;

    // Sieve of Eratosthenes
    var prime = new Array(maxVal + 1);
    prime.fill(true);
    sieve(prime, maxVal);

    // To store the required sum
    var sum = 0;

    for (var i = 2; i <= maxVal; i++) {

        // If current element is prime
        if (prime[i]) {
            sum += i;
        }
    }

    return sum;
}

var d = 8;
 document.write(  sumPrime(d));



//This code is contributed by SoumikMondal
</script>

Output

6081

Time Complexity: O(sqrt(2^d))
Auxiliary Space: O(2^d)