给定整数n和基数B ,任务是找到n!的长度!在基地B中。
例子:
Input: n = 4, b = 10
Output: 2
Explanation: 4! = 24, hence number of digits is 2
Input: n = 4, b = 16
Output: 2
Explanation: 4! = 18 in base 16, hence number of digits is 2
方法:
为了解决该问题,我们使用Kamenetsky公式,该公式近似计算阶乘中的位数
f(x) = log10( ((n/e)^n) * sqrt(2*pi*n))
n中以b为底的位数由logb(n)= log10(n)/ log10(b)给出。因此,通过使用对数的性质,可以通过以下方法获得基数b中阶乘的位数:
f(x) = ( n* log10(( n/ e)) + log10(2*pi*n)/2 ) / log10(b)
这种方法可以处理可容纳在32位整数甚至更大的整数中的大型输入!
下面的代码是上述想法的实现:
C++
// A optimised program to find the
// number of digits in a factorial in base b
#include
using namespace std;
// Returns the number of digits present
// in n! in base b Since the result can be large
// long long is used as return type
long long findDigits(int n, int b)
{
// factorial of -ve number
// doesn't exists
if (n < 0)
return 0;
// base case
if (n <= 1)
return 1;
// Use Kamenetsky formula to calculate
// the number of digits
double x = ((n * log10(n / M_E) +
log10(2 * M_PI * n) /
2.0)) / (log10(b));
return floor(x) + 1;
}
// Driver Code
int main()
{
//calling findDigits(Number, Base)
cout << findDigits(4, 16) << endl;
cout << findDigits(5, 8) << endl;
cout << findDigits(12, 16) << endl;
cout << findDigits(19, 13) << endl;
return 0;
}
Java
// A optimised program to find the
// number of digits in a factorial in base b
class GFG{
// Returns the number of digits present
// in n! in base b Since the result can be large
// long is used as return type
static long findDigits(int n, int b)
{
// factorial of -ve number
// doesn't exists
if (n < 0)
return 0;
// base case
if (n <= 1)
return 1;
double M_PI = 3.141592;
double M_E = 2.7182;
// Use Kamenetsky formula to calculate
// the number of digits
double x = ((n * Math.log10(n / M_E) +
Math.log10(2 * M_PI * n) /
2.0)) / (Math.log10(b));
return (long) (Math.floor(x) + 1);
}
// Driver Code
public static void main(String[] args)
{
//calling findDigits(Number, Base)
System.out.print(findDigits(4, 16) +"\n");
System.out.print(findDigits(5, 8) +"\n");
System.out.print(findDigits(12, 16) +"\n");
System.out.print(findDigits(19, 13) +"\n");
}
}
// This code is contributed by 29AjayKumar
Python 3
from math import log10,floor
# A optimised program to find the
# number of digits in a factorial in base b
# Returns the number of digits present
# in n! in base b Since the result can be large
# long long is used as return type
def findDigits(n, b):
# factorial of -ve number
# doesn't exists
if (n < 0):
return 0
M_PI = 3.141592
M_E = 2.7182
# base case
if (n <= 1):
return 1
# Use Kamenetsky formula to calculate
# the number of digits
x = ((n * log10(n / M_E) + log10(2 * M_PI * n) / 2.0)) / (log10(b))
return floor(x) + 1
# Driver Code
if __name__ == '__main__':
#calling findDigits(Number, Base)
print(findDigits(4, 16))
print(findDigits(5, 8))
print(findDigits(12, 16))
print(findDigits(19, 13))
# This code is contributed by Surendra_Gangwar
C#
// A optimised C# program to find the
// number of digits in a factorial in base b
using System;
class GFG{
// Returns the number of digits present
// in n! in base b Since the result can be large
// long is used as return type
static long findDigits(int n, int b)
{
// factorial of -ve number
// doesn't exists
if (n < 0)
return 0;
// base case
if (n <= 1)
return 1;
double M_PI = 3.141592;
double M_E = 2.7182;
// Use Kamenetsky formula to calculate
// the number of digits
double x = ((n * Math.Log10(n / M_E) +
Math.Log10(2 * M_PI * n) /
2.0)) / (Math.Log10(b));
return (long) (Math.Floor(x) + 1);
}
// Driver Code
public static void Main(string[] args)
{
// calling findDigits(Number, Base)
Console.WriteLine(findDigits(4, 16));
Console.WriteLine(findDigits(5, 8));
Console.WriteLine(findDigits(12, 16));
Console.WriteLine(findDigits(19, 13));
}
}
// This code is contributed by Yash_R
Javascript
输出:
2
3
8
16
参考: oeis.org