Gegeven een getal n is het de taak om de primoriaal ervan te berekenen. Primoriaal (aangeduid als PN#) is een product van de eerste n priemgetallen. Primitief van een getal is vergelijkbaar met de faculteit van een getal. In het primoriaal worden niet alle natuurlijke getallen vermenigvuldigd. Alleen priemgetallen worden vermenigvuldigd om de primoriaal van een getal te berekenen. Het wordt aangegeven met P#.
Voorbeelden:
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 naïeve aanpak is om alle getallen van 1 tot n één voor één te controleren of ze een priemgetal zijn of niet. Zo ja, sla dan de vermenigvuldiging op in het resultaat en sla op dezelfde manier het resultaat op van de vermenigvuldiging van priemgetallen tot n.
Een efficiënt methode is om alle priemgetallen tot en met n te vinden met behulp van Zeef van Sundaram en bereken dan gewoon de primoriaal door ze allemaal te vermenigvuldigen.
C++
// C++ program to find Primorial of given numbers #include 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 // 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 # 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(1int((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(1int(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(1n+1): print('Primorial(P#) of'i'is'calculatePrimorial(i)); # This code is contributed by mits
C# // 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 ?> JavaScript <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) + '
'); // This code is contributed by gfgking </script>
Uitgang:
grafiek voor toewijzing van middelen
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
Tijdcomplexiteit: - O(N)