Given a number n, the task is to calculate its primorial. Primorial (denoted as Pn#) is a product of first n prime numbers. Primorial of a number is similar to the factorial of a number. In primorial, not all the natural numbers get multiplied only prime numbers are multiplied to calculate the primorial of a number. It is denoted with P#.
Examples:
Input: n = 3
Output: 30
Primorial = 2 * 3 * 5 = 30
As a side note, factorial is 2 * 3 * 4 * 5
Input: n = 5
Output: 2310
Primorial = 2 * 3 * 5 * 7 * 11
A naive approach is to check all numbers from 1 to n one by one is prime or not, if yes then store the multiplication in result, similarly store the result of multiplication of primes till n.
An efficient method is to find all the prime up-to n using Sieve of Sundaram and then just calculate the primorial by multiplying them all.
C++
// C++ program to find Primorial of given numbers
#include<bits/stdc++.h>
using namespace std;
const int MAX = 1000000;
// vector to store all prime less than and equal to 10^6
vector <int> primes;
// Function for sieve of sundaram. This function stores all
// prime numbers less than MAX in primes
void sieveSundaram()
{
// In general Sieve of Sundaram, produces primes smaller
// than (2*x + 2) for a number given number x. Since
// we want primes smaller than MAX, we reduce MAX to half
// This array is used to separate numbers of the form
// i+j+2ij from others where 1 <= i <= j
bool marked[MAX/2 + 1] = {0};
// Main logic of Sundaram. Mark all numbers which
// do not generate prime number by doing 2*i+1
for (int i = 1; i <= (sqrt(MAX)-1)/2 ; i++)
for (int j = (i*(i+1))<<1 ; j <= MAX/2 ; j += 2*i +1)
marked[j] = true;
// Since 2 is a prime number
primes.push_back(2);
// Print other primes. Remaining primes are of the
// form 2*i + 1 such that marked[i] is false.
for (int i=1; i<=MAX/2; i++)
if (marked[i] == false)
primes.push_back(2*i + 1);
}
// Function to calculate primorial of n
int calculatePrimorial(int n)
{
// Multiply first n primes
int result = 1;
for (int i=0; i<n; i++)
result = result * primes[i];
return result;
}
// Driver code
int main()
{
int n = 5;
sieveSundaram();
for (int i = 1 ; i<= n; i++)
cout << "Primorial(P#) of " << i << " is "
<< calculatePrimorial(i) <<endl;
return 0;
}
// Java program to find Primorial of given numbers
import java.util.*;
class GFG{
public static int MAX = 1000000;
// vector to store all prime less than and equal to 10^6
static ArrayList<Integer> primes = new ArrayList<Integer>();
// Function for sieve of sundaram. This function stores all
// prime numbers less than MAX in primes
static void sieveSundaram()
{
// In general Sieve of Sundaram, produces primes smaller
// than (2*x + 2) for a number given number x. Since
// we want primes smaller than MAX, we reduce MAX to half
// This array is used to separate numbers of the form
// i+j+2ij from others where 1 <= i <= j
boolean[] marked = new boolean[MAX];
// Main logic of Sundaram. Mark all numbers which
// do not generate prime number by doing 2*i+1
for (int i = 1; i <= (Math.sqrt(MAX) - 1) / 2 ; i++)
{
for (int j = (i * (i + 1)) << 1 ; j <= MAX / 2 ; j += 2 * i + 1)
{
marked[j] = true;
}
}
// Since 2 is a prime number
primes.add(2);
// Print other primes. Remaining primes are of the
// form 2*i + 1 such that marked[i] is false.
for (int i = 1; i <= MAX / 2; i++)
{
if (marked[i] == false)
{
primes.add(2 * i + 1);
}
}
}
// Function to calculate primorial of n
static int calculatePrimorial(int n)
{
// Multiply first n primes
int result = 1;
for (int i = 0; i < n; i++)
{
result = result * primes.get(i);
}
return result;
}
// Driver code
public static void main(String[] args)
{
int n = 5;
sieveSundaram();
for (int i = 1 ; i <= n; i++)
{
System.out.println("Primorial(P#) of "+i+" is "+calculatePrimorial(i));
}
}
}
// This Code is contributed by mits
# Python3 program to find Primorial of given numbers
import math
MAX = 1000000;
# vector to store all prime less than and equal to 10^6
primes=[];
# Function for sieve of sundaram. This function stores all
# prime numbers less than MAX in primes
def sieveSundaram():
# In general Sieve of Sundaram, produces primes smaller
# than (2*x + 2) for a number given number x. Since
# we want primes smaller than MAX, we reduce MAX to half
# This array is used to separate numbers of the form
# i+j+2ij from others where 1 <= i <= j
marked=[False]*(int(MAX/2)+1);
# Main logic of Sundaram. Mark all numbers which
# do not generate prime number by doing 2*i+1
for i in range(1,int((math.sqrt(MAX)-1)/2)+1):
for j in range(((i*(i+1))<<1),(int(MAX/2)+1),(2*i+1)):
marked[j] = True;
# Since 2 is a prime number
primes.append(2);
# Print other primes. Remaining primes are of the
# form 2*i + 1 such that marked[i] is false.
for i in range(1,int(MAX/2)):
if (marked[i] == False):
primes.append(2*i + 1);
# Function to calculate primorial of n
def calculatePrimorial(n):
# Multiply first n primes
result = 1;
for i in range(n):
result = result * primes[i];
return result;
# Driver code
n = 5;
sieveSundaram();
for i in range(1,n+1):
print("Primorial(P#) of",i,"is",calculatePrimorial(i));
# This code is contributed by mits
// C# program to find Primorial of given numbers
using System;
using System.Collections;
class GFG{
public static int MAX = 1000000;
// vector to store all prime less than and equal to 10^6
static ArrayList primes = new ArrayList();
// Function for sieve of sundaram. This function stores all
// prime numbers less than MAX in primes
static void sieveSundaram()
{
// In general Sieve of Sundaram, produces primes smaller
// than (2*x + 2) for a number given number x. Since
// we want primes smaller than MAX, we reduce MAX to half
// This array is used to separate numbers of the form
// i+j+2ij from others where 1 <= i <= j
bool[] marked = new bool[MAX];
// Main logic of Sundaram. Mark all numbers which
// do not generate prime number by doing 2*i+1
for (int i = 1; i <= (Math.Sqrt(MAX) - 1) / 2 ; i++)
{
for (int j = (i * (i + 1)) << 1 ; j <= MAX / 2 ; j += 2 * i + 1)
{
marked[j] = true;
}
}
// Since 2 is a prime number
primes.Add(2);
// Print other primes. Remaining primes are of the
// form 2*i + 1 such that marked[i] is false.
for (int i = 1; i <= MAX / 2; i++)
{
if (marked[i] == false)
{
primes.Add(2 * i + 1);
}
}
}
// Function to calculate primorial of n
static int calculatePrimorial(int n)
{
// Multiply first n primes
int result = 1;
for (int i = 0; i < n; i++)
{
result = result * (int)primes[i];
}
return result;
}
// Driver code
public static void Main()
{
int n = 5;
sieveSundaram();
for (int i = 1 ; i <= n; i++)
{
System.Console.WriteLine("Primorial(P#) of "+i+" is "+calculatePrimorial(i));
}
}
}
// This Code is contributed by mits
<?php
// PHP program to find Primorial
// of given numbers
$MAX = 100000;
// vector to store all prime less
// than and equal to 10^6
$primes = array();
// Function for sieve of sundaram.
// This function stores all prime
// numbers less than MAX in primes
function sieveSundaram()
{
global $MAX, $primes;
// In general Sieve of Sundaram,
// produces primes smaller than
// (2*x + 2) for a number given
// number x. Since we want primes
// smaller than MAX, we reduce MAX
// to half. This array is used to
// separate numbers of the form
// i+j+2ij from others where 1 <= i <= j
$marked = array_fill(0, $MAX / 2 + 1, 0);
// Main logic of Sundaram. Mark all numbers which
// do not generate prime number by doing 2*i+1
for ($i = 1; $i <= (sqrt($MAX) - 1) / 2 ; $i++)
for ($j = ($i * ($i + 1)) << 1 ;
$j <= $MAX / 2 ; $j += 2 * $i + 1)
$marked[$j] = true;
// Since 2 is a prime number
array_push($primes, 2);
// Print other primes. Remaining primes
// are of the form 2*i + 1 such that
// marked[i] is false.
for ($i = 1; $i <= $MAX / 2; $i++)
if ($marked[$i] == false)
array_push($primes, (2 * $i + 1));
}
// Function to calculate primorial of n
function calculatePrimorial($n)
{
global $primes;
// Multiply first n primes
$result = 1;
for ($i = 0; $i < $n; $i++)
$result = $result * $primes[$i];
return $result;
}
// Driver code
$n = 5;
sieveSundaram();
for ($i = 1 ; $i<= $n; $i++)
echo "Primorial(P#) of " . $i .
" is " . calculatePrimorial($i) . "\n";
// This code is contributed by mits
?>
<script>
// Javascript program to find Primorial
// of given numbers
let MAX = 100000;
// vector to store all prime less
// than and equal to 10^6
let primes = new Array();
// Function for sieve of sundaram.
// This function stores all prime
// numbers less than MAX in primes
function sieveSundaram()
{
// In general Sieve of Sundaram,
// produces primes smaller than
// (2*x + 2) for a number given
// number x. Since we want primes
// smaller than MAX, we reduce MAX
// to half. This array is used to
// separate numbers of the form
// i+j+2ij from others where 1 <= i <= j
let marked = new Array(MAX / 2 + 1).fill(0);
// Main logic of Sundaram. Mark all numbers which
// do not generate prime number by doing 2*i+1
for (let i = 1; i <= (Math.sqrt(MAX) - 1) / 2 ; i++)
for (let j = (i * (i + 1)) << 1 ;
j <= MAX / 2 ; j += 2 * i + 1)
marked[j] = true;
// Since 2 is a prime number
primes.push(2);
// Print other primes. Remaining primes
// are of the form 2*i + 1 such that
// marked[i] is false.
for (let i = 1; i <= MAX / 2; i++)
if (marked[i] == false)
primes.push(2 * i + 1);
}
// Function to calculate primorial of n
function calculatePrimorial(n)
{
// Multiply first n primes
let result = 1;
for (let i = 0; i < n; i++)
result = result * primes[i];
return result;
}
// Driver code
let n = 5;
sieveSundaram();
for (let i = 1 ; i<= n; i++)
document.write("Primorial(P#) of " + i + " is " +
calculatePrimorial(i) + "<br>");
// This code is contributed by gfgking
</script>
Output:
Primorial(P#) of 1 is 2
Primorial(P#) of 2 is 6
Primorial(P#) of 3 is 30
Primorial(P#) of 4 is 210
Primorial(P#) of 5 is 2310
Time complexity :- O(N)
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