Étant donné un nombre n, la tâche consiste à calculer sa primitive. Primorial (noté Pn#) est un produit des n premiers nombres premiers. Primorial d'un nombre est similaire à la factorielle d'un nombre. En primorial, tous les nombres naturels ne sont pas multipliés, seuls les nombres premiers sont multipliés pour calculer le primorial d'un nombre. Il est noté P#.
Exemples :
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
UN approche naïve est de vérifier que tous les nombres de 1 à n un par un sont premiers ou non, si oui, puis de stocker la multiplication dans le résultat, de stocker de la même manière le résultat de la multiplication des nombres premiers jusqu'à n.
Un efficace La méthode consiste à trouver tous les nombres premiers jusqu'à n en utilisant Tamis de Sundaram puis calculez simplement la primitive en les multipliant toutes.
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>
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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
Complexité temporelle :- O(N)