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Bir sayının primali

Bir n sayısı verildiğinde görev onun ilkselini hesaplamaktır. İlkel (P olarak gösterilirN#) ilk n asal sayının çarpımıdır. İlkel Bir sayının faktöriyeli, bir sayının faktöriyeline benzer. İlkelde tüm doğal sayılar çarpılmaz, bir sayının ilkelini hesaplamak için yalnızca asal sayılar çarpılır. P# ile gösterilir.
Örnekler:  
 

  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 


 

Önerilen Uygulama Bir sayının primali Deneyin!


A naif yaklaşım 1'den n'ye kadar tüm sayıları tek tek kontrol etmek, asal olup olmadığını kontrol etmek, eğer evet ise, çarpmayı sonuçta saklamak, benzer şekilde n'ye kadar asal sayıların çarpımının sonucunu saklamaktır.
Bir verimli yöntemi kullanarak n'ye kadar olan tüm asal sayıları bulmaktır. Sundaram Eleği ve sonra hepsini çarparak ilkseli hesaplayın.
 



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>

Çıkış:   

mason formülü
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

Zaman karmaşıklığı :-  O(N) 


 

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