Dado un número n, imprima todos los números primos menores o iguales que n.
Ejemplos:
Input: n = 10 Output: 2, 3, 5, 7 Input: n = 20 Output: 2, 3, 5, 7, 11, 13, 17, 19
Hemos discutido el algoritmo Tamiz de Eratóstenes para la tarea anterior.
A continuación se muestra el algoritmo Tamiz de Sundaram.
printPrimes(n) [Prints all prime numbers smaller than n] 1) In general Sieve of Sundaram, produces primes smaller than (2*x + 2) for a number given number x. Since we want primes smaller than n, we reduce n-1 to half. We call it nNew. nNew = (n-1)/2; For example, if n = 102, then nNew = 50. if n = 103, then nNew = 51 2) Create an array marked[n] that is going to be used to separate numbers of the form i+j+2ij from others where 1 <= i <= j 3) Initialize all entries of marked[] as false. 4) // Mark all numbers of the form i + j + 2ij as true // where 1 <= i <= j Loop for i=1 to nNew a) j = i; b) Loop While (i + j + 2*i*j) 2, then print 2 as first prime. 6) Remaining primes are of the form 2i + 1 where i is index of NOT marked numbers. So print 2i + 1 for all i such that marked[i] is false.
A continuación se muestra la implementación del algoritmo anterior:
C++
// C++ program to print primes smaller than n using // Sieve of Sundaram. #include <bits/stdc++.h> using namespace std; // Prints all prime numbers smaller int SieveOfSundaram(int n) { // In general Sieve of Sundaram, produces primes smaller // than (2*x + 2) for a number given number x. // Since we want primes smaller than n, we reduce n to half int nNew = (n-1)/2; // This array is used to separate numbers of the form i+j+2ij // from others where 1 <= i <= j bool marked[nNew + 1]; // Initialize all elements as not marked memset(marked, false, sizeof(marked)); // Main logic of Sundaram. Mark all numbers of the // form i + j + 2ij as true where 1 <= i <= j for (int i=1; i<=nNew; i++) for (int j=i; (i + j + 2*i*j) <= nNew; j++) marked[i + j + 2*i*j] = true; // Since 2 is a prime number if (n > 2) cout << 2 << " "; // Print other primes. Remaining primes are of the form // 2*i + 1 such that marked[i] is false. for (int i=1; i<=nNew; i++) if (marked[i] == false) cout << 2*i + 1 << " "; } // Driver program to test above int main(void) { int n = 20; SieveOfSundaram(n); return 0; }
Java
// Java program to print primes smaller // than n using Sieve of Sundaram. import java.util.Arrays; class GFG { // Prints all prime numbers smaller static int SieveOfSundaram(int n) { // In general Sieve of Sundaram, produces // primes smaller than (2*x + 2) for a number // given number x. Since we want primes // smaller than n, we reduce n to half int nNew = (n - 1) / 2; // This array is used to separate numbers of the // form i+j+2ij from others where 1 <= i <= j boolean marked[] = new boolean[nNew + 1]; // Initialize all elements as not marked Arrays.fill(marked, false); // Main logic of Sundaram. Mark all numbers of the // form i + j + 2ij as true where 1 <= i <= j for (int i = 1; i <= nNew; i++) for (int j = i; (i + j + 2 * i * j) <= nNew; j++) marked[i + j + 2 * i * j] = true; // Since 2 is a prime number if (n > 2) System.out.print(2 + " "); // Print other primes. Remaining primes are of // the form 2*i + 1 such that marked[i] is false. for (int i = 1; i <= nNew; i++) if (marked[i] == false) System.out.print(2 * i + 1 + " "); return -1; } // Driver code public static void main(String[] args) { int n = 20; SieveOfSundaram(n); } } // This code is contributed by Anant Agarwal.
Python3
# Python3 program to print # primes smaller than n using # Sieve of Sundaram. # Prints all prime numbers smaller def SieveOfSundaram(n): # In general Sieve of Sundaram, # produces primes smaller # than (2*x + 2) for a number # given number x. Since we want # primes smaller than n, we # reduce n to half nNew = int((n - 1) / 2); # This array is used to separate # numbers of the form i+j+2ij # from others where 1 <= i <= j # Initialize all elements as not marked marked = [0] * (nNew + 1); # Main logic of Sundaram. Mark all # numbers of the form i + j + 2ij # as true where 1 <= i <= j for i in range(1, nNew + 1): j = i; while((i + j + 2 * i * j) <= nNew): marked[i + j + 2 * i * j] = 1; j += 1; # Since 2 is a prime number if (n > 2): print(2, end = " "); # Print other primes. Remaining # primes are of the form 2*i + 1 # such that marked[i] is false. for i in range(1, nNew + 1): if (marked[i] == 0): print((2 * i + 1), end = " "); # Driver Code n = 20; SieveOfSundaram(n); # This code is contributed by mits
C#
// C# program to print primes smaller // than n using Sieve of Sundaram. using System; class GFG { // Prints all prime numbers smaller static int SieveOfSundaram(int n) { // In general Sieve of Sundaram, produces // primes smaller than (2*x + 2) for a number // given number x. Since we want primes // smaller than n, we reduce n to half int nNew = (n - 1) / 2; // This array is used to separate // numbers of the form i+j+2ij from // others where 1 <= i <= j bool []marked = new bool[nNew + 1]; // Initialize all elements as not marked for (int i=0;i<nNew+1;i++) marked[i]=false; // Main logic of Sundaram. // Mark all numbers of the // form i + j + 2ij as true // where 1 <= i <= j for (int i = 1; i <= nNew; i++) for (int j = i; (i + j + 2 * i * j) <= nNew; j++) marked[i + j + 2 * i * j] = true; // Since 2 is a prime number if (n > 2) Console.Write(2 + " "); // Print other primes. // Remaining primes are of // the form 2*i + 1 such // that marked[i] is false. for (int i = 1; i <= nNew; i++) if (marked[i] == false) Console.Write(2 * i + 1 + " "); return -1; } // Driver code public static void Main() { int n = 20; SieveOfSundaram(n); } } // This code is contributed by nitin mittal
PHP
<?php // PHP program to print primes smaller // than n using Sieve of Sundaram. // Prints all prime numbers smaller function SieveOfSundaram($n) { // In general Sieve of Sundaram, // produces primes smaller than // (2*x + 2) for a number given // number x. Since we want primes // smaller than n, we reduce n to half $nNew = ($n - 1) / 2; // This array is used to separate // numbers of the form i+j+2ij // from others where 1 <= i <= j // Initialize all elements as not marked $marked = array_fill(0, ($nNew + 1), false); // Main logic of Sundaram. Mark all // numbers of the form i + j + 2ij // as true where 1 <= i <= j for ($i = 1; $i <= $nNew; $i++) for ($j = $i; ($i + $j + 2 * $i * $j) <= $nNew; $j++) $marked[$i + $j + 2 * $i * $j] = true; // Since 2 is a prime number if ($n > 2) echo "2 "; // Print other primes. Remaining // primes are of the form 2*i + 1 // such that marked[i] is false. for ($i = 1; $i <= $nNew; $i++) if ($marked[$i] == false) echo (2 * $i + 1) . " "; } // Driver Code $n = 20; SieveOfSundaram($n); // This code is contributed by mits ?>
Javascript
<script> // JavaScript program to print primes smaller // than n using Sieve of Sundaram. // Prints all prime numbers smaller function SieveOfSundaram(n) { // In general Sieve of Sundaram, produces // primes smaller than (2*x + 2) for a number // given number x. Since we want primes // smaller than n, we reduce n to half let nNew = (n - 1) / 2; // This array is used to separate // numbers of the form i+j+2ij from // others where 1 <= i <= j let marked = []; // Initialize all elements as not marked for (let i = 0; i < nNew + 1; i++) marked[i] = false; // Main logic of Sundaram. // Mark all numbers of the // form i + j + 2ij as true // where 1 <= i <= j for (let i = 1; i <= nNew; i++) for (let j = i; (i + j + 2 * i * j) <= nNew; j++) marked[i + j + 2 * i * j] = true; // Since 2 is a prime number if (n > 2) document.write(2 + " "); // Print other primes. // Remaining primes are of // the form 2*i + 1 such // that marked[i] is false. for (let i = 1; i <= nNew; i++) if (marked[i] == false) document.write(2 * i + 1 + " "); return -1; } // Driver program let n = 20; SieveOfSundaram(n); // This code is contributed by susmitakundugoaldanga. </script>
2 3 5 7 11 13 17 19
Ilustración:
Todas las entradas rojas en la siguiente ilustración son entradas marcadas. Para cada entrada restante (o negra) x, el número 2x+1 es primo.
Veamos como funciona para n=102, tendremos el tamiz para (n-1)/2 de la siguiente manera:
Marque todos los números que se pueden representar como i + j + 2ij
Ahora, para todos los números sin marcar en la lista, encuentra 2x+1 y ese será el primo:
Como 2*1+1=3
2*3+1=7
2*5+1=11
2*6+1=13
2*8+1=17 y así sucesivamente…
¿Cómo funciona esto?
Cuando producimos nuestro resultado final, producimos todos los enteros de la forma 2x+1 (es decir, son impares) excepto 2, que se maneja por separado.
Let q be an integer of the form 2x + 1. q is excluded if and only if x is of the form i + j + 2ij. That means, q = 2(i + j + 2ij) + 1 = (2i + 1)(2j + 1) So, an odd integer is excluded from the final list if and only if it has a factorization of the form (2i + 1)(2j + 1) which is to say, if it has a non-trivial odd factor. Source: Wiki
Referencia:
https://en.wikipedia.org/wiki/Sieve_of_Sundaram
Este artículo es una contribución de Anuj Rathore . Escriba comentarios si encuentra algo incorrecto o si desea compartir más información sobre el tema tratado anteriormente.
Publicación traducida automáticamente
Artículo escrito por GeeksforGeeks-1 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA