给定由两个正整数L和R表示的范围。任务是对具有素数幂的GCD等于1的范围内的数进行计数。换句话说，如果数X的素数分解形式为2 p 1 * 3 p 2 * 5 p 3 *…，则p 1 ，p 2 ，p 3 …的GCD应该等于1 。
例子：
Input: L = 2, R = 5
Output: 3
2, 3, and 5 are the required numbers having GCD of powers of prime factors equal to 1.
2 = 21
3 = 31
5 = 51
Input: L = 13, R = 20
Output: 7
先决条件：在一定范围内拥有完美的力量
天真的方法：迭代从L到R的所有数字，并对每个数字进行素数分解，然后计算素数的幂的GCD。如果GCD = 1 ，则增加一个计数变量，最后返回它作为答案。
高效方法：此处的关键思想是要注意有效数不是完美幂，因为质数因子的幂总是使它们的GCD始终大于1。换句话说，所有完美幂都不是有效数。
例如
2500 is perfect power whose prime factorization is 2500 = 22 * 54. Now the GCD of (2, 4) = 2 which is greater than 1.
If some number has xth power of a factor in its prime factorization, then the powers of other prime factors will have to be multiples of x in order for the number to be invalid.
因此，我们可以找到处于该范围内的理想功效的总数，并将其从总数中减去。
下面是上述方法的实现：
CPP
// C++ implementation of the approach
#include
using namespace std;
#define N 1000005
#define MAX 1e18
// Vector to store powers greater than 3
vector powers;
// Set to store perfect squares
set squares;
// Set to store powers other than perfect squares
set s;
void powersPrecomputation()
{
for (long int i = 2; i < N; i++) {
// Pushing squares
squares.insert(i * i);
// if the values is already a perfect square means
// present in the set
if (squares.find(i) != squares.end())
continue;
long int temp = i;
// Run loop until some power of current number
// doesn't exceed MAX
while (i * i <= MAX / temp) {
temp *= (i * i);
// Pushing only odd powers as even power of a number
// can always be expressed as a perfect square
// which is already present in set squares
s.insert(temp);
}
}
// Inserting those sorted
// values of set into a vector
for (auto x : s)
powers.push_back(x);
}
long int calculateAnswer(long int L, long int R)
{
// Precompute the powers
powersPrecomputation();
// Calculate perfect squares in
// range using sqrtl function
long int perfectSquares = floor(sqrtl(R)) - floor(sqrtl(L - 1));
// Calculate upper value of R
// in vector using binary search
long int high = upper_bound(powers.begin(),
powers.end(), R)
- powers.begin();
// Calculate lower value of L
// in vector using binary search
long int low = lower_bound(powers.begin(),
powers.end(), L)
- powers.begin();
// Calculate perfect powers
long perfectPowers = perfectSquares + (high - low);
// Compute final answer
long ans = (R - L + 1) - perfectPowers;
return ans;
}
// Driver Code
int main()
{
long int L = 13, R = 20;
cout << calculateAnswer(L, R);
return 0;
} Java
// Java implementation of above idea
import java.util.*;
class GFG
{
static int N = 1000005;
static long MAX = (long) 1e18;
// Vector to store powers greater than 3
static Vector powers = new Vector<>();
// Set to store perfect squares
static TreeSet squares = new TreeSet<>();
// Set to store powers other than perfect squares
static TreeSet s = new TreeSet<>();
static void powersPrecomputation()
{
for (long i = 2; i < N; i++)
{
// Pushing squares
squares.add(i * i);
// if the values is already a perfect square means
// present in the set
if (squares.contains(i))
continue;
long temp = i;
// Run loop until some power of current number
// doesn't exceed MAX
while (i * i <= MAX / temp)
{
temp *= (i * i);
// Pushing only odd powers as even power of a number
// can always be expressed as a perfect square
// which is already present in set squares
s.add(temp);
}
}
// Inserting those sorted
// values of set into a vector
for (long x : s)
powers.add(x);
}
static long calculateAnswer(long L, long R)
{
// Precompute the powers
powersPrecomputation();
// Calculate perfect squares in
// range using sqrtl function
long perfectSquares = (long) (Math.floor(Math.sqrt(R)) -
Math.floor(Math.sqrt(L - 1)));
// Calculate upper value of R
// in vector using binary search
long high = Collections.binarySearch(powers, R);
// Calculate lower value of L
// in vector using binary search
long low = Collections.binarySearch(powers, L);
// Calculate perfect powers
long perfectPowers = perfectSquares + (high - low);
// Compute final answer
long ans = (R - L + 1) - perfectPowers;
return ans;
}
// Driver Code
public static void main(String[] args)
{
long L = 13, R = 20;
System.out.println(calculateAnswer(L, R));
}
}
// This code is contributed by
// sanjeev2552 Python3
# Python3 implementation of the approach
from bisect import bisect as upper_bound
from bisect import bisect_left as lower_bound
from math import floor
N = 1000005
MAX = 10**18
# Vector to store powers greater than 3
powers = []
# Set to store perfect squares
squares = dict()
# Set to store powers other than perfect squares
s = dict()
def powersPrecomputation():
for i in range(2, N):
# Pushing squares
squares[i * i] = 1
# if the values is already a perfect square means
# present in the set
if (i not in squares.keys()):
continue
temp = i
# Run loop until some power of current number
# doesn't exceed MAX
while (i * i <= (MAX // temp)):
temp *= (i * i)
# Pushing only odd powers as even power of a number
# can always be expressed as a perfect square
# which is already present in set squares
s[temp]=1
# Inserting those sorted
# values of set into a vector
for x in s:
powers.append(x)
def calculateAnswer(L, R):
# Precompute the powers
powersPrecomputation()
# Calculate perfect squares in
# range using sqrtl function
perfectSquares = floor((R)**(.5)) - floor((L - 1)**(.5))
# Calculate upper value of R
# in vector using binary search
high = upper_bound(powers,R)
# Calculate lower value of L
# in vector using binary search
low = lower_bound(powers,L)
# Calculate perfect powers
perfectPowers = perfectSquares + (high - low)
# Compute final answer
ans = (R - L + 1) - perfectPowers
return ans
# Driver Code
L = 13
R = 20
print(calculateAnswer(L, R))
# This code is contributed by mohit kumar 29C#
// C# implementation of above idea
using System;
using System.Collections.Generic;
public class GFG
{
static int N = 100005;
static long MAX = (long) 1e18;
// List to store powers greater than 3
static List powers = new List();
// Set to store perfect squares
static HashSet squares = new HashSet();
// Set to store powers other than perfect squares
static HashSet s = new HashSet();
static void powersPrecomputation()
{
for (long i = 2; i < N; i++)
{
// Pushing squares
squares.Add(i * i);
// if the values is already a perfect square means
// present in the set
if (squares.Contains(i))
continue;
long temp = i;
// Run loop until some power of current number
// doesn't exceed MAX
while (i * i <= MAX / temp)
{
temp *= (i * i);
// Pushing only odd powers as even power of a number
// can always be expressed as a perfect square
// which is already present in set squares
s.Add(temp);
}
}
// Inserting those sorted
// values of set into a vector
foreach (long x in s)
powers.Add(x);
}
static long calculateAnswer(long L, long R)
{
// Precompute the powers
powersPrecomputation();
// Calculate perfect squares in
// range using sqrtl function
long perfectSquares = (long) (Math.Floor(Math.Sqrt(R)) -
Math.Floor(Math.Sqrt(L - 1)));
// Calculate upper value of R
// in vector using binary search
long high = Array.BinarySearch(powers.ToArray(), R);
// Calculate lower value of L
// in vector using binary search
long low = Array.BinarySearch(powers.ToArray(), L);
// Calculate perfect powers
long perfectPowers = perfectSquares + (high - low);
// Compute readonly answer
long ans = (R - L + 1) - perfectPowers;
return ans;
}
// Driver Code
public static void Main(String[] args)
{
long L = 13, R = 20;
Console.WriteLine(calculateAnswer(L, R));
}
}
// This code is contributed by 29AjayKumar 输出：
7