Check if an integer can be expressed as a sum of two semi-primes

Given a positive integer N, check if it can be expressed as a sum of two semi-primes or not.

Semi-primes A number is said to be a semi-prime if it can be expressed as product of two primes number ( not necessarily distinct ).

Semi-primes in the range 1 -100 are:

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95.

Examples:

Input : N = 30
Output: YES
Explanation: 30 can be expressed as '15 + 15' 
15 is semi-primes as 15 is a product of two primes 3 and 5.

Input : N = 45
Output : YES
Explanation: 45 can be expressed as '35 + 10'
35 and 10 are also  semi-primes as it can a be expressed 
as product of two primes:
       35 = 5 * 7
       10 = 2 * 5

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Prerequisite:

Semi-primes

A Simple Solution is to traverse form i=1 to $N$ and check if i and N-i are semi-primes or not. If yes, print i and n-i.

An Efficient Solution is to pre-compute semi-primes in an array up to the given range and then traverse the semi-prime array and check if n-arr[i] is semi-prime or not. As, arr[i] is already a semi-prime if n-arr[i] is also a semi-prime, then n can be expressed as sum of two semi-primes.

Below is the implementation of above approach:

C++

// CPP Code to check if an integer 
// can be expressed as sum of 
// two semi-primes 
  
#include <bits/stdc++.h> 
using namespace std; 
#define MAX 1000000 
  
vector<int> arr; 
bool sprime[MAX]; 
  
// Utility function to compute 
// semi-primes in a range 
void computeSemiPrime() 
{ 
    memset(sprime, false, sizeof(sprime)); 
  
    for (int i = 2; i < MAX; i++) { 
  
        int cnt = 0; 
        int num = i; 
        for (int j = 2; cnt < 2 && j * j <= num; ++j) { 
            while (num % j == 0) { 
                num /= j, ++cnt; // Increment count 
                // of prime numbers 
            } 
        } 
  
        // If number is greater than 1, add it to 
        // the count variable as it indicates the 
        // number remain is prime number 
  
        if (num > 1) 
            ++cnt; 
  
        // if count is equal to '2' then 
        // number is semi-prime 
  
        if (cnt == 2) { 
  
            sprime[i] = true; 
            arr.push_back(i); 
        } 
    } 
} 
  
// Utility function to check 
// if a number sum of two 
// semi-primes 
bool checkSemiPrime(int n) 
{ 
    int i = 0; 
  
    while (arr[i] <= n / 2) { 
  
        // arr[i] is already a semi-prime 
        // if n-arr[i] is also a semi-prime 
        // then we a number can be expressed as 
        // sum of two semi-primes 
  
        if (sprime[n - arr[i]]) { 
            return true; 
        } 
  
        i++; 
    } 
  
    return false; 
} 
  
// Driver code 
int main() 
{ 
    computeSemiPrime(); 
  
    int n = 30; 
    if (checkSemiPrime(n)) 
        cout << "YES"; 
    else
        cout << "NO"; 
  
    return 0; 
} 

Java

// Java Code to check if an integer 
// can be expressed as sum of 
// two semi-primes 
  
import java.util.*; 
  
class GFG { 
  
    static final int MAX = 1000000; 
    static Vector<Integer> arr = new Vector<>(); 
    static boolean[] sprime = new boolean[MAX]; 
  
    // Utility function to compute 
    // semi-primes in a range 
    static void computeSemiPrime() 
    { 
  
        for (int i = 0; i < MAX; i++) 
            sprime[i] = false; 
  
        for (int i = 2; i < MAX; i++) { 
  
            int cnt = 0; 
            int num = i; 
            for (int j = 2; cnt < 2 && j * j <= num; ++j) { 
                while (num % j == 0) { 
                    num /= j; 
                    ++cnt; 
                    // Increment count 
                    // of prime numbers 
                } 
            } 
  
            // If number is greater than 1, add it to 
            // the count variable as it indicates the 
            // number remain is prime number 
  
            if (num > 1) 
                ++cnt; 
  
            // if count is equal to '2' then 
            // number is semi-prime 
  
            if (cnt == 2) { 
  
                sprime[i] = true; 
                arr.add(i); 
            } 
        } 
    } 
  
    // Utility function to check 
    // if a number is sum of two 
    // semi-primes 
    static boolean checkSemiPrime(int n) 
    { 
        int i = 0; 
  
        while (arr.get(i) <= n / 2) { 
  
            // arr[i] is already a semi-prime 
            // if n-arr[i] is also a semi-prime 
            // then we a number can be expressed as 
            // sum of two semi-primes 
  
            if (sprime[n - arr.get(i)]) { 
                return true; 
            } 
  
            i++; 
        } 
  
        return false; 
    } 
  
    // Driver code 
    public static void main(String[] args) 
    { 
        computeSemiPrime(); 
  
        int n = 30; 
        if (checkSemiPrime(n)) 
            System.out.println("YES"); 
        else
            System.out.println("NO"); 
    } 
} 

Python3

# Python3 Code to check if an integer can  
# be expressed as sum of two semi-primes  
MAX = 10000
  
arr = []  
sprime = [False] * (MAX)  
  
# Utility function to compute  
# semi-primes in a range  
def computeSemiPrime(): 
  
    for i in range(2, MAX):  
  
        cnt, num, j = 0, i, 2
        while cnt < 2 and j * j <= num:  
            while num % j == 0:  
                num /= j  
                  
                # Increment count of prime numbers 
                cnt += 1
                  
            j += 1
  
        # If number is greater than 1, add it  
        # to the count variable as it indicates  
        # the number remain is prime number  
        if num > 1: 
            cnt += 1
  
        # if count is equal to '2' then  
        # number is semi-prime  
        if cnt == 2: 
  
            sprime[i] = True
            arr.append(i)  
  
# Utility function to check  
# if a number sum of two  
# semi-primes  
def checkSemiPrime(n):  
  
    i = 0
    while arr[i] <= n // 2: 
  
        # arr[i] is already a semi-prime  
        # if n-arr[i] is also a semi-prime  
        # then we a number can be expressed as  
        # sum of two semi-primes  
        if sprime[n - arr[i]] == True: 
            return True
  
        i += 1
      
    return False
  
# Driver code  
if __name__ == "__main__":  
  
    computeSemiPrime()  
  
    n = 30
    if checkSemiPrime(n) == True:  
        print("YES")  
    else: 
        print("NO") 
  
# This code is contributed by  
# Rituraj Jain 

C#

// C# Code to check if an integer 
// can be expressed as sum of 
// two semi-primes 
using System.Collections.Generic; 
  
class GFG  
{ 
  
static int MAX = 1000000; 
static List<int> arr = new List<int>(); 
static bool[] sprime = new bool[MAX]; 
  
// Utility function to compute 
// semi-primes in a range 
static void computeSemiPrime() 
{ 
  
    for (int i = 0; i < MAX; i++) 
        sprime[i] = false; 
  
    for (int i = 2; i < MAX; i++)  
    { 
  
        int cnt = 0; 
        int num = i; 
        for (int j = 2; cnt < 2 && j * j <= num; ++j) 
        { 
            while (num % j == 0)  
            { 
                num /= j; 
                ++cnt; 
                  
                // Increment count 
                // of prime numbers 
            } 
        } 
  
        // If number is greater than 1, add it to 
        // the count variable as it indicates the 
        // number remain is prime number 
        if (num > 1) 
            ++cnt; 
  
        // if count is equal to '2' then 
        // number is semi-prime 
  
        if (cnt == 2)  
        { 
            sprime[i] = true; 
            arr.Add(i); 
        } 
    } 
} 
  
// Utility function to check 
// if a number is sum of two 
// semi-primes 
static bool checkSemiPrime(int n) 
{ 
    int i = 0; 
  
    while (arr[i] <= n / 2)  
    { 
  
        // arr[i] is already a semi-prime 
        // if n-arr[i] is also a semi-prime 
        // then we a number can be expressed as 
        // sum of two semi-primes 
  
        if (sprime[n - arr[i]])  
        { 
            return true; 
        } 
  
        i++; 
    } 
  
    return false; 
} 
  
// Driver code 
public static void Main() 
{ 
    computeSemiPrime(); 
  
    int n = 30; 
    if (checkSemiPrime(n)) 
        System.Console.WriteLine("YES"); 
    else
        System.Console.WriteLine("NO"); 
} 
} 
  
// This code is contributed by mits 

Output:

YES

My Personal Notes

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python3

C#

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