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Find the number of ordered pairs such that a * p + b * q = N, where p and q are primes
  • Last Updated : 08 Apr, 2021

Given an array arr[], and integer Q denoting number of queries and two numbers a, b, the task is to find the number of ordered pairs (p, q) such that a * p + b * q = arr[i], where p and q are prime numbers. 
Examples: 
 

Input: Q = 3, arr[] = { 2, 7, 11 }, a = 1, b = 2 
Output: 0 1 2 
Explanation: 
2 -> There are no ordered pairs (p, q) such that p + 2*q = 2. 
7 -> There is only one ordered pair (p, q) = (3, 2) such that p + 2*q = 7. 
11 -> There are two ordered pairs (p, q) = (7, 2), (5, 3) such that p + 2*q = 11.
Input: Q = 2, arr[] = { 15, 25 }, a = 1, b = 2 
Output: 2 3
 

 

Approach: The idea is to store every prime number in an array using Sieve of Eratosthenes. After storing the prime numbers, count the number of ordered pairs (p, q) such that a*p + b*q = N for every combination of (p, q) in the prime array.
Below is the implementation of the above approach:
 

C++




// C++ program to find the number of ordered
// pairs such that a * p + b * q = N
// where p and q are primes
 
#include <bits/stdc++.h>
#define size 10001
using namespace std;
int prime[size];
int freq[size];
 
// Sieve of erastothenes
// to store the prime numbers
// and their frequency in form a*p+b*q
void sieve(int a, int b)
{
    prime[1] = 1;
 
    // Performing Sieve of Eratosthenes
    // to find the prime numbers unto 10001
    for (int i = 2; i * i < size; i++) {
        if (prime[i] == 0) {
            for (int j = i * 2; j < size; j += i)
                prime[j] = 1;
        }
    }
 
    // Loop to find the number of
    // ordered pairs for every combination
    // of the prime numbers
    for (int p = 1; p < size; p++) {
        for (int q = 1; q < size; q++) {
            if (prime[p] == 0 && prime[q] == 0
                && a * p + b * q < size) {
                freq[a * p + b * q]++;
            }
        }
    }
}
 
// Driver code
int main()
{
    int queries = 2, a = 1, b = 2;
    sieve(a, b);
    int arr[queries] = { 15, 25 };
 
    // Printing the number of ordered pairs
    // for every query
    for (int i = 0; i < queries; i++) {
        cout << freq[arr[i]] << " ";
    }
 
    return 0;
}

Java




// Java program to find the number of ordered
// pairs such that a * p + b * q = N
// where p and q are primes
public class GFG {
 
   
    final static int size = 10001;
    static int prime[] = new int[size];
    static int freq[] = new int [size];
     
    // Sieve of erastothenes
    // to store the prime numbers
    // and their frequency in form a*p+b*q
    static void sieve(int a, int b)
    {
        prime[1] = 1;
     
        // Performing Sieve of Eratosthenes
        // to find the prime numbers unto 10001
        for (int i = 2; i * i < size; i++) {
            if (prime[i] == 0) {
                for (int j = i * 2; j < size; j += i)
                    prime[j] = 1;
            }
        }
     
        // Loop to find the number of
        // ordered pairs for every combination
        // of the prime numbers
        for (int p = 1; p < size; p++) {
            for (int q = 1; q < size; q++) {
                if (prime[p] == 0 && prime[q] == 0
                    && a * p + b * q < size) {
                    freq[a * p + b * q]++;
                }
            }
        }
    }
     
    // Driver code
    public static void main (String[] args)
    {
        int queries = 2, a = 1, b = 2;
        sieve(a, b);
        int arr[] = { 15, 25 };
     
        // Printing the number of ordered pairs
        // for every query
        for (int i = 0; i < queries; i++) {
            System.out.print(freq[arr[i]] + " ");
        }
     
     
    }
}
// This code is contributed by AnkitRai01

Python3




# Python3 program to find the number of ordered
# pairs such that a * p + b * q = N
# where p and q are primes
from math import sqrt
size = 1000
prime = [0 for i in range(size)]
freq = [0 for i in range(size)]
 
# Sieve of erastothenes
# to store the prime numbers
# and their frequency in form a*p+b*q
def sieve(a, b):
    prime[1] = 1
 
    # Performing Sieve of Eratosthenes
    # to find the prime numbers unto 10001
    for i in range(2, int(sqrt(size)) + 1, 1):
        if (prime[i] == 0):
            for j in range(i*2, size, i):
                prime[j] = 1
 
    # Loop to find the number of
    # ordered pairs for every combination
    # of the prime numbers
    for p in range(1, size, 1):
        for q in range(1, size, 1):
            if (prime[p] == 0 and prime[q] == 0 and a * p + b * q < size):
                freq[a * p + b * q] += 1
 
# Driver code
if __name__ == '__main__':
    queries = 2
    a = 1
    b = 2
    sieve(a, b)
    arr = [15, 25]
 
    # Printing the number of ordered pairs
    # for every query
    for i in range(queries):
        print(freq[arr[i]],end = " ")
 
# This code is contributed by Surendra_Gangwar

C#




// C# program to find the number of ordered
// pairs such that a * p + b * q = N
// where p and q are primes
using System;
 
class GFG {
 
     
    static int size = 10001;
    static int []prime = new int[size];
    static int []freq = new int [size];
     
    // Sieve of erastothenes
    // to store the prime numbers
    // and their frequency in form a*p+b*q
    static void sieve(int a, int b)
    {
        prime[1] = 1;
     
        // Performing Sieve of Eratosthenes
        // to find the prime numbers unto 10001
        for (int i = 2; i * i < size; i++) {
            if (prime[i] == 0) {
                for (int j = i * 2; j < size; j += i)
                    prime[j] = 1;
            }
        }
     
        // Loop to find the number of
        // ordered pairs for every combination
        // of the prime numbers
        for (int p = 1; p < size; p++) {
            for (int q = 1; q < size; q++) {
                if (prime[p] == 0 && prime[q] == 0
                    && a * p + b * q < size) {
                    freq[a * p + b * q]++;
                }
            }
        }
    }
     
    // Driver code
    public static void Main (string[] args)
    {
        int queries = 2, a = 1, b = 2;
        sieve(a, b);
        int []arr = { 15, 25 };
     
        // Printing the number of ordered pairs
        // for every query
        for (int i = 0; i < queries; i++) {
            Console.Write(freq[arr[i]] + " ");
        }
     
     
    }
}
 
// This code is contributed by AnkitRai01

Javascript




<script>
 
// JavaScript program to find the number of ordered
// pairs such that a * p + b * q = N
// where p and q are primes
 
size=10001
prime = Array(size).fill(0);
freq = Array(size).fill(0);
 
// Sieve of erastothenes
// to store the prime numbers
// and their frequency in form a*p+b*q
function sieve(a, b)
{
    prime[1] = 1;
 
    // Performing Sieve of Eratosthenes
    // to find the prime numbers unto 10001
    for (var i = 2; i * i < size; i++) {
        if (prime[i] == 0) {
            for (var j = i * 2; j < size; j += i)
                prime[j] = 1;
        }
    }
 
    // Loop to find the number of
    // ordered pairs for every combination
    // of the prime numbers
    for (var p = 1; p < size; p++) {
        for (var q = 1; q < size; q++) {
            if (prime[p] == 0 && prime[q] == 0
                && a * p + b * q < size) {
                freq[a * p + b * q]++;
            }
        }
    }
}
 
// Driver code
var queries = 2, a = 1, b = 2;
sieve(a, b);
var arr = [ 15, 25 ];
 
// Printing the number of ordered pairs
// for every query
for(var i = 0; i < queries; i++) {
    document.write(freq[arr[i]] + " ");
}
 
</script>
Output: 
2 3

 

Time Complexity: O(N)
 

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