P – smooth numbers in given ranges
Given multiple ranges [L, R] and a prime number p, we need to find all P-smooth numbers in given individual ranges.
What is P – smooth number? An integer is P – smooth number if the largest Prime factor of that number <= p. 1 is considered (by OEIS) as P – smooth number for any possible value of P because it does not have any prime factor.
Examples:
Input : p = 7
ranges[] = {[1, 17], [10, 25]}
Output :
For first range : 1 2 3 4 5 6 7 8 9 12 14 15 16
For second range : 15 16 18 20 21 24 25
Explanation : Largest prime factors of numbers
printed above are less than or equal to 7.
Suppose, we are checking 7 – smooth numbers. 1. Consider an integer 56. Here, 56 = 2 * 2 * 2 * 7. So, 56 has two prime factors (2 and 7) which are <=7. So, 56 is 7-smooth number. 2. Consider another integer 66. Here, 66 = 2 * 3 * 11. 66 has three prime factors (2, 3 and 11). Where 11>7. So 66 is not 7-smooth number. Brute – Force Approach: Let P and range [L, R] is given. Here L <= R. Create a loop and check for all numbers in inclusive range [L : R]. If that number has largest prime factor <= p. Then print that number (i.e. P-smooth number). Calculate its Largest Prime Factor / Divisor, using maxPrimeDivisor(n) function.
Efficient Approach: The idea is to pre-compute p-smooth numbers for maximum value of all ranges. Once we have pre-computed, we can quickly print for all ranges one by one.
C++
// C++ program to display p-smooth// number in given range.// P-smooth numbers' array#include <bits/stdc++.h>using namespace std;vector<int> p_smooth;// Returns Maximum Prime// Divisor of nint maxPrimeDivisor(int n){ int MPD = -1; if (n == 1) { return 1; } while (n % 2 == 0) { MPD = 2; n = n / 2; } // math.sqrt(n) + 1 int size = pow(n, 0.5) + 1; for (int odd = 3; odd < size; odd += 2) { while (n % odd == 0) { // Make sure no multiples // of prime, enters here MPD = odd; n /= odd; } } // When n is prime itself MPD = max(n, MPD); return MPD;}// generates p-smooth numbers.void generate_p_smooth(int p, int MAX_LIMIT){ for (int i = 2; i < MAX_LIMIT + 1; i++) { if (maxPrimeDivisor(i) <= p) { // Satisfies the condition // of p-smooth number p_smooth.push_back(i); } }}// finds p-smooth number in the// given [L:R] range.void find_p_smooth(int L, int R){ if (L <= p_smooth[p_smooth.size() - 1]) { // If user input exceeds MAX_LIMIT // range, no checking for (int i = 0; i < p_smooth.size(); i++) { int w = p_smooth[i]; if (w > R) break; if (w >= L && w <= R) { // Print P-smooth numbers // within range : L to R. cout << w << " "; } } cout << "\n"; }}// Driver Codeint main(){ p_smooth.push_back(1); // p_smooth number : p = 7 // L <= R int p = 7; int L = 1; int R = 100; // Maximum possible value of R int MAX_LIMIT = 1000; // generate the p-smooth numbers generate_p_smooth(p, MAX_LIMIT); // Find an print the p-smooth numbers find_p_smooth(L, R);}// The code is contributed by phasing17 |
Java
// Java program to display p-smooth// number in given range.// P-smooth numbers' arrayimport java.util.*;class GFG { static ArrayList<Integer> p_smooth = new ArrayList<Integer>(); // Returns Maximum Prime // Divisor of n static int maxPrimeDivisor(int n) { int MPD = -1; if (n == 1) { return 1; } while (n % 2 == 0) { MPD = 2; n = n / 2; } // math.sqrt(n) + 1 int size = (int)Math.pow(n, 0.5) + 1; for (int odd = 3; odd < size; odd += 2) { while (n % odd == 0) { // Make sure no multiples // of prime, enters here MPD = odd; n /= odd; } } // When n is prime itself MPD = Math.max(n, MPD); return MPD; } // generates p-smooth numbers. static void generate_p_smooth(int p, int MAX_LIMIT) { for (int i = 2; i < MAX_LIMIT + 1; i++) { if (maxPrimeDivisor(i) <= p) { // Satisfies the condition // of p-smooth number p_smooth.add(i); } } } // finds p-smooth number in the // given [L:R] range. static void find_p_smooth(int L, int R) { if (L <= p_smooth.get(p_smooth.size() - 1)) { // If user input exceeds MAX_LIMIT // range, no checking for (int i = 0; i < p_smooth.size(); i++) { int w = p_smooth.get(i); if (w > R) break; if (w >= L && w <= R) { // Print P-smooth numbers // within range : L to R. System.out.print(w + " "); } } System.out.print("\n"); } } // Driver Code public static void main(String[] args) { p_smooth.add(1); // p_smooth number : p = 7 // L <= R int p = 7; int L = 1; int R = 100; // Maximum possible value of R int MAX_LIMIT = 1000; // generate the p-smooth numbers generate_p_smooth(p, MAX_LIMIT); // Find an print the p-smooth numbers find_p_smooth(L, R); }}// The code is contributed by phasing17 |
Python3
# Python program to display p-smooth# number in given range.# P-smooth numbers' arrayp_smooth = [1] def maxPrimeDivisor(n): # Returns Maximum Prime # Divisor of n MPD = -1 if n == 1 : return 1 while n % 2 == 0: MPD = 2 n = n // 2 # math.sqrt(n) + 1 size = int(n ** 0.5) + 1 for odd in range( 3, size, 2 ): while n % odd == 0: # Make sure no multiples # of prime, enters here MPD = odd n = n // odd # When n is prime itself MPD = max (n, MPD) return MPD def generate_p_smooth(p, MAX_LIMIT): # generates p-smooth numbers. global p_smooth for i in range(2, MAX_LIMIT + 1): if maxPrimeDivisor(i) <= p: # Satisfies the condition # of p-smooth number p_smooth.append(i) def find_p_smooth(L, R): # finds p-smooth number in the # given [L:R] range. global p_smooth if L <= p_smooth[-1]: # If user input exceeds MAX_LIMIT # range, no checking for w in p_smooth : if w > R : break if w >= L and w <= R : # Print P-smooth numbers # within range : L to R. print(w, end =" ") print() # p_smooth number : p = 7# L <= Rp = 7L, R = 1, 100 # Maximum possible value of RMAX_LIMIT = 1000 # generate the p-smooth numbersgenerate_p_smooth(p, MAX_LIMIT) # Find an print the p-smooth numbersfind_p_smooth(L, R) |
C#
// C# program to display p-smooth// number in given range.// P-smooth numbers' arrayusing System;using System.Collections.Generic;class GFG { static List<int> p_smooth = new List<int>(); // Returns Maximum Prime // Divisor of n static int maxPrimeDivisor(int n) { int MPD = -1; if (n == 1) { return 1; } while (n % 2 == 0) { MPD = 2; n = n / 2; } // math.sqrt(n) + 1 int size = (int)Math.Pow(n, 0.5) + 1; for (int odd = 3; odd < size; odd += 2) { while (n % odd == 0) { // Make sure no multiples // of prime, enters here MPD = odd; n /= odd; } } // When n is prime itself MPD = Math.Max(n, MPD); return MPD; } // generates p-smooth numbers. static void generate_p_smooth(int p, int MAX_LIMIT) { for (int i = 2; i < MAX_LIMIT + 1; i++) { if (maxPrimeDivisor(i) <= p) { // Satisfies the condition // of p-smooth number p_smooth.Add(i); } } } // finds p-smooth number in the // given [L:R] range. static void find_p_smooth(int L, int R) { if (L <= p_smooth[p_smooth.Count - 1]) { // If user input exceeds MAX_LIMIT // range, no checking for (int i = 0; i < p_smooth.Count; i++) { int w = p_smooth[i]; if (w > R) break; if (w >= L && w <= R) { // Print P-smooth numbers // within range : L to R. Console.Write(w + " "); } } Console.Write("\n"); } } // Driver Code public static void Main(string[] args) { p_smooth.Add(1); // p_smooth number : p = 7 // L <= R int p = 7; int L = 1; int R = 100; // Maximum possible value of R int MAX_LIMIT = 1000; // generate the p-smooth numbers generate_p_smooth(p, MAX_LIMIT); // Find an print the p-smooth numbers find_p_smooth(L, R); }}// The code is contributed by phasing17 |
Javascript
// JavaScript program to display p-smooth// number in given range.// P-smooth numbers' arraylet p_smooth = [1];// Returns Maximum Prime// Divisor of nfunction maxPrimeDivisor(n){ let MPD = -1; if (n == 1){ return 1; } while(n % 2 == 0){ MPD = 2; n = Math.floor(n/2); } // math.sqrt(n) + 1 let size = Math.floor(n ** 0.5) + 1; for(let odd = 3; odd < size; odd += 2){ while(n % odd == 0){ // Make sure no multiples // of prime, enters here MPD = odd; n = Math.floor(n/odd); } } // When n is prime itself MPD = Math.max(n, MPD); return MPD;}// generates p-smooth numbers.function generate_p_smooth(p, MAX_LIMIT){ for(let i = 2; i < MAX_LIMIT+1; i++){ if(maxPrimeDivisor(i) <= p){ //Satisfies the condition // of p-smooth number p_smooth.push(i); } }} // finds p-smooth number in the// given [L:R] range.function find_p_smooth(L, R){ if(L <= p_smooth[p_smooth.length-1]){ // If user input exceeds MAX_LIMIT // range, no checking for(let i = 0; i < p_smooth.length; i++){ let w = p_smooth[i]; if (w > R) break; if (w >= L && w <= R){ // Print P-smooth numbers // within range : L to R. document.write(w + " "); } } document.write("\n"); }} // p_smooth number : p = 7// L <= Rlet p = 7;let L = 1;let R = 100;// Maximum possible value of Rlet MAX_LIMIT = 1000// generate the p-smooth numbersgenerate_p_smooth(p, MAX_LIMIT)// Find an print the p-smooth numbersfind_p_smooth(L, R)// The code is contributed by Gautam goel (gautamgoel962) |
Output: 1 2 3 4 5 6 7 8 9 10 12 14 15 16 18 20 21 24 25 27 28 30 32 35 36 40 42 45 48 49 50 54 56 60 63 64 70 72 75 80 81 84 90 96 98 100
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