最大素因数大于其平方根的 N 之前的最大整数

给定一个正整数N ，任务是找到[1, N]范围内的最大数，使得该数的平方根小于其最大素因数。

Input: N = 15
Output: 15
Explanation: The prime factors of 15 are {3, 5}. The square root of 15 is 3.87 (i.e, 3.87 < 5). Therefore 15 is the largest valid integer in the given range.

Input: N = 25
Output: 23

方法：给定的问题可以通过使用埃拉托色尼筛进行一些修改来解决。创建一个数组gpf[] ，它存储给定范围内所有整数的最大素因子。最初， gpf[] = {0} 。使用 Sieve，将数组gpf[]的所有索引初始化为相应索引的最大素因子，类似于本文中讨论的算法。

现在，以相反的方式迭代范围[N, 1]并打印其平方根小于其最大素因数的第一个整数。

下面是上述方法的实现：

C++

// C++ program for the above approach
 
#include 
using namespace std;
 
const int maxn = 100001;
 
// Stores the Greatest Prime Factor
int gpf[maxn];
 
// Modified Sieve to find the Greatest
// Prime Factor of all integers in the
// range [1, maxn]
void modifiedSieve()
{
    // Initialize the array with 0
    memset(gpf, 0, sizeof(gpf));
    gpf[0] = 0;
    gpf[1] = 1;
 
    // Iterate through all values of i
    for (int i = 2; i < maxn; i++) {
 
        // If i is not a prime number
        if (gpf[i] > 0)
            continue;
 
        // Update the multiples of i
        for (int j = i; j < maxn; j += i) {
            gpf[j] = max(i, gpf[j]);
        }
    }
}
 
// Function to find integer in the range
// [1, N] such that its Greatest Prime
// factor is greater than its square root
int greatestValidInt(int N)
{
 
    modifiedSieve();
 
    // Iterate through all values of
    // i in the range [N, 1]
    for (int i = N; i > 0; i--) {
 
        // If greatest prime factor of i
        // is greater than its square root
        if (gpf[i] > sqrt(i)) {
 
            // Return answer
            return i;
        }
    }
 
    // If no valid integer exist
    return -1;
}
 
// Driver Code
int main()
{
    int N = 25;
    cout << greatestValidInt(N);
 
    return 0;
}

Java

// Java program for the above approach
public class GFG {
     
    final static int maxn = 100001;
     
    // Stores the Greatest Prime Factor
    static int gpf[] = new int[maxn];
     
    // Modified Sieve to find the Greatest
    // Prime Factor of all integers in the
    // range [1, maxn]
    static void modifiedSieve()
    {
       
        // Initialize the array with 0
        for (int i = 0; i < maxn; i++ )
            gpf[i] = 0;
             
        gpf[0] = 0;
        gpf[1] = 1;
     
        // Iterate through all values of i
        for (int i = 2; i < maxn; i++) {
     
            // If i is not a prime number
            if (gpf[i] > 0)
                continue;
     
            // Update the multiples of i
            for (int j = i; j < maxn; j += i) {
                gpf[j] = Math.max(i, gpf[j]);
            }
        }
    }
     
    // Function to find integer in the range
    // [1, N] such that its Greatest Prime
    // factor is greater than its square root
    static int greatestValidInt(int N)
    {
     
        modifiedSieve();
     
        // Iterate through all values of
        // i in the range [N, 1]
        for (int i = N; i > 0; i--) {
     
            // If greatest prime factor of i
            // is greater than its square root
            if (gpf[i] > Math.sqrt(i)) {
     
                // Return answer
                return i;
            }
        }
     
        // If no valid integer exist
        return -1;
    }
     
    // Driver Code
    public static void main (String[] args)
    {
        int N = 25;
        System.out.println(greatestValidInt(N));
    }
}
 
// This code is contributed by AnkThon

Python3

# python program for the above approach
 
import math
 
maxn = 100001
 
# Stores the Greatest Prime Factor
gpf = [0 for _ in range(maxn)]
 
# Modified Sieve to find the Greatest
# Prime Factor of all integers in the
# range [1, maxn]
 
 
def modifiedSieve():
 
    # Initialize the array with 0
    gpf[0] = 0
    gpf[1] = 1
 
    # Iterate through all values of i
    for i in range(2, maxn):
 
        # If i is not a prime number
        if (gpf[i] > 0):
            continue
 
        # Update the multiples of i
        for j in range(i, maxn, i):
            gpf[j] = max(i, gpf[j])
 
 
# Function to find integer in the range
# [1, N] such that its Greatest Prime
# factor is greater than its square root
def greatestValidInt(N):
 
    modifiedSieve()
 
    # Iterate through all values of
    # i in the range [N, 1]
    for i in range(N, 0, -1):
 
        # If greatest prime factor of i
        # is greater than its square root
        if (gpf[i] > math.sqrt(i)):
 
            # Return answer
            return i
 
    # If no valid integer exist
    return -1
 
 
# Driver Code
if __name__ == "__main__":
 
    N = 25
    print(greatestValidInt(N))
 
# This code is contributed by rakeshsahni

C#

// C# program for the above approach
using System;
public class GFG {
 
    static int maxn = 100001;
 
    // Stores the Greatest Prime Factor
    static int[] gpf = new int[maxn];
 
    // Modified Sieve to find the Greatest
    // Prime Factor of all integers in the
    // range [1, maxn]
    static void modifiedSieve()
    {
 
        // Initialize the array with 0
        for (int i = 0; i < maxn; i++)
            gpf[i] = 0;
 
        gpf[0] = 0;
        gpf[1] = 1;
 
        // Iterate through all values of i
        for (int i = 2; i < maxn; i++) {
 
            // If i is not a prime number
            if (gpf[i] > 0)
                continue;
 
            // Update the multiples of i
            for (int j = i; j < maxn; j += i) {
                gpf[j] = Math.Max(i, gpf[j]);
            }
        }
    }
 
    // Function to find integer in the range
    // [1, N] such that its Greatest Prime
    // factor is greater than its square root
    static int greatestValidInt(int N)
    {
 
        modifiedSieve();
 
        // Iterate through all values of
        // i in the range [N, 1]
        for (int i = N; i > 0; i--) {
 
            // If greatest prime factor of i
            // is greater than its square root
            if (gpf[i] > Math.Sqrt(i)) {
 
                // Return answer
                return i;
            }
        }
 
        // If no valid integer exist
        return -1;
    }
 
    // Driver Code
    public static void Main(string[] args)
    {
        int N = 25;
        Console.WriteLine(greatestValidInt(N));
    }
}
 
// This code is contributed by ukasp.

Javascript

// Javascript program for the above approach
 
const maxn = 100001;
 
// Stores the Greatest Prime Factor
let gpf = new Array(maxn);
 
// Modified Sieve to find the Greatest
// Prime Factor of all integers in the
// range [1, maxn]
function modifiedSieve()
{
 
  // Initialize the array with 0
  gpf.fill(0);
  gpf[0] = 0;
  gpf[1] = 1;
 
  // Iterate through all values of i
  for (let i = 2; i < maxn; i++)
  {
   
    // If i is not a prime number
    if (gpf[i] > 0) continue;
 
    // Update the multiples of i
    for (let j = i; j < maxn; j += i) {
      gpf[j] = Math.max(i, gpf[j]);
    }
  }
}
 
// Function to find integer in the range
// [1, N] such that its Greatest Prime
// factor is greater than its square root
function greatestValidInt(N) {
  modifiedSieve();
 
  // Iterate through all values of
  // i in the range [N, 1]
  for (let i = N; i > 0; i--)
  {
   
    // If greatest prime factor of i
    // is greater than its square root
    if (gpf[i] > Math.sqrt(i))
    {
     
      // Return answer
      return i;
    }
  }
 
  // If no valid integer exist
  return -1;
}
 
// Driver Code
let N = 25;
document.write(greatestValidInt(N));
 
// This code is contributed by gfgking.

输出：

23

时间复杂度： O(N*log N)
辅助空间： O(1)