Donat un nombre n, la tasca és calcular-ne el primorial. Primorial (indicat com a Pn#) és un producte dels n primers nombres primers. Primordial d'un nombre és semblant al factorial d'un nombre. En el primorial no es multipliquen tots els nombres naturals, només es multipliquen els nombres primers per calcular el primorial d'un nombre. Es denota amb 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
A enfocament ingenu és comprovar que tots els nombres de l'1 al n un per un siguin primers o no, en cas afirmatiu, emmagatzemeu la multiplicació en el resultat de la mateixa manera emmagatzemeu el resultat de la multiplicació de nombres primers fins a n.
An eficient El mètode és trobar tots els primers fins a n utilitzant Tamís de Sundaram i després només cal calcular el primorial multiplicant-los tots.
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
Complexitat temporal:- O(N)