Mobius Function
Examples:
Input : 6 Output : 1 Solution: Prime Factors: 2 3. Therefore p = 2, (-1)^p = 1 Input: 49 Output: 0 Solution: Prime Factors: 7 ( occurs twice). Since the prime factor occurs twice answer is 0. Input: 3 Output: -1 Solution: Prime Factors: 3. Therefore p = 1, (-1) ^ p =-1 Input : 78 Output : 1 Solution: Prime Factors: 3, 13. Therefore p = 2, (-1)^p = 1
Method 1 (Simple)
We iterate through all numbers i smaller than or equal to N. For every number we check if it divides N. If yes, we check if it’s also prime. If both conditions are satisfied, we check if its square also divides N. If yes, we return 0. If the square doesn’t divide, we increment count of prime factors. Finally, we return 1 if there are an even number of prime factors and return -1 if there are odd number of prime factors.
// Java program for mobius function import java.io.*;
public class GFG {
// C# Program to evaluate Mobius
// Function: M(N) = 1 if N = 1
// M(N) = 0 if any prime factor
// of N is contained twice
// M(N) = (-1)^(no of distinct
// prime factors)
// Function to check if n is
// prime or not
static boolean isPrime( int n)
{
if (n < 2 )
return false ;
for ( int i = 2 ; i * i <= n; i++)
if (n % i == 0 )
return false ;
return true ;
}
static int mobius( int N)
{
// Base Case
if (N == 1 )
return 1 ;
// For a prime factor i check if
// i^2 is also a factor.
int p = 0 ;
for ( int i = 1 ; i <= N; i++) {
if (N % i == 0 && isPrime(i)) {
// Check if N is divisible by i^2
if (N % (i * i) == 0 )
return 0 ;
else
// i occurs only once, increase p
p++;
}
}
// All prime factors are contained only
// once Return 1 if p is even else -1
return (p % 2 != 0 ) ? - 1 : 1 ;
}
// Driver code
static public void main(String[] args)
{
int N = 17 ;
System.out.println( "Mobius Functions M(N) at " +
" N = " + N + " is: " + mobius(N));
System.out.println( "Mobius Functions M(N) at " +
" N = " + 25 + " is: " + mobius( 25 ));
System.out.println( "Mobius Functions M(N) at " +
" N = " + 6 + " is: " + mobius( 6 ));
}
} // This code is contributed by vt_m |
// C# Program to evaluate Mobius Function using System;
public class GFG
{ // M(N) = 1 if N = 1
// M(N) = 0 if any prime factor
// of N is contained twice
// M(N) = (-1)^(no of distinct
// prime factors)
// Function to check if n is
// prime or not
static bool isPrime( int n)
{
if (n == 2)
return true ;
if (n % 2 == 0)
return false ;
for ( int i = 3; i * i <= n / 2; i += 2)
if (n % i == 0)
return false ;
return true ;
}
static int mobius( int N)
{
// Base Case
if (N == 1)
return 1;
// For a prime factor i check
// if i^2 is also a factor.
int p = 0;
for ( int i = 2; i <= N; i++)
{
if (N % i == 0 && isPrime(i)) {
// Check if N is divisible by i^2
if (N % (i * i) == 0)
return 0;
else
// i occurs only once, increase p
p++;
}
}
// All prime factors are contained only
// once Return 1 if p is even else -1
return (p % 2 != 0) ? -1 : 1;
}
// Driver code
static public void Main()
{
Console.WriteLine( "Mobius Functions M(N) at " +
"N = " + 17 + " is: " + mobius(17));
Console.WriteLine( "Mobius Functions M(N) at " +
"N = " + 25 + " is: " + mobius(25));
Console.WriteLine( "Mobius Functions M(N) at " +
"N = " + 6 + " is: " + mobius(6));
}
} // This code is contributed by vt_m |
<?php // PHP Program to evaluate Mobius Function // M(N) = 1 if N = 1 // M(N) = 0 if any prime factor of // N is contained twice // M(N) = (-1)^(no of distinct prime factors) // Function to check if n is prime or not function isPrime( $n )
{ if ( $n < 2)
return false;
for ( $i = 2; $i * $i <= $n ; $i ++)
if ( $n % $i == 0)
return false;
return true;
} function mobius( $N )
{ // Base Case
if ( $N == 1)
return 1;
// For a prime factor i
// check if i^2 is also
// a factor.
$p = 0;
for ( $i = 1; $i <= $N ; $i ++) {
if ( $N % $i == 0 && isPrime( $i )) {
// Check if N is divisible by i^2
if ( $N % ( $i * $i ) == 0)
return 0;
else
// i occurs only once, increase p
$p ++;
}
}
// All prime factors are
// contained only once
// Return 1 if p is even
// else -1
return ( $p % 2 != 0) ? -1 : 1;
} // Driver Code
$N = 17;
echo "Mobius Functions M(N) at N = " , $N , " is: "
, mobius( $N ) , "\n" ;
echo "Mobius Functions M(N) at N = " ,25, " is: "
, mobius(25), "\n" ;
echo "Mobius Functions M(N) at N = " ,6, " is: "
, mobius(6) ;
// This code is contributed by nitin mittal. ?> |
<script> // JavaScript Program to evaluate Mobius
// Function: M(N) = 1 if N = 1
// M(N) = 0 if any prime factor
// of N is contained twice
// M(N) = (-1)^(no of distinct
// prime factors)
// Function to check if n is
// prime or not
function isPrime(n)
{
if (n < 2)
return false ;
for (let i = 2; i * i <= n; i++)
if (n % i == 0)
return false ;
return true ;
}
function mobius(N)
{
// Base Case
if (N == 1)
return 1;
// For a prime factor i check if
// i^2 is also a factor.
let p = 0;
for (let i = 1; i <= N; i++) {
if (N % i == 0 && isPrime(i)) {
// Check if N is divisible by i^2
if (N % (i * i) == 0)
return 0;
else
// i occurs only once, increase p
p++;
}
}
// All prime factors are contained only
// once Return 1 if p is even else -1
return (p % 2 != 0) ? -1 : 1;
}
// Driver code let N = 17;
document.write( "Mobius Functions M(N) at " +
" N = " + N + " is: " + mobius(N) + "<br/>" );
document.write( "Mobius Functions M(N) at " +
" N = " + 25 + " is: " + mobius(25) + "<br/>" );
document.write( "Mobius Functions M(N) at " +
" N = " + 6 + " is: " + mobius(6) + "<br/>" );
</script> |
Output:
Mobius Functions M(N) at N = 17 is: -1 Mobius Functions M(N) at N = 25 is: 0 Mobius Functions M(N) at N = 6 is: 1
Time Complexity: O(n?n )
Auxiliary Space: O(1)
Method 2 (Efficient)
The idea is based on efficient program to print all prime factors of a given number. The interesting thing is, we do not need inner while loop here because if a number divides more than once, we can immediately return 0.
// Program to print all prime factors # include <bits/stdc++.h> using namespace std;
// Returns value of mobius() int mobius( int n)
{ int p = 0;
// Handling 2 separately
if (n%2 == 0)
{
n = n/2;
p++;
// If 2^2 also divides N
if (n % 2 == 0)
return 0;
}
// Check for all other prime factors
for ( int i = 3; i <= sqrt (n); i = i+2)
{
// If i divides n
if (n%i == 0)
{
n = n/i;
p++;
// If i^2 also divides N
if (n % i == 0)
return 0;
}
}
return (p % 2 == 0)? -1 : 1;
} // Driver code int main()
{
int N = 17;
cout << "Mobius Functions M(N) at N = " << N << " is: "
<< mobius(N) << endl;
cout << "Mobius Functions M(N) at N = " << 25 << " is: "
<< mobius(25) << endl;
cout << "Mobius Functions M(N) at N = " << 6 << " is: "
<< mobius(6) << endl;
} |
// Java program to print all prime factors import java.io.*;
class GFG {
// Returns value of mobius()
static int mobius( int n)
{
int p = 0 ;
// Handling 2 separately
if (n % 2 == 0 )
{
n = n / 2 ;
p++;
// If 2^2 also divides N
if (n % 2 == 0 )
return 0 ;
}
// Check for all other prime factors
for ( int i = 3 ; i <= Math.sqrt(n);
i = i+ 2 )
{
// If i divides n
if (n % i == 0 )
{
n = n / i;
p++;
// If i^2 also divides N
if (n % i == 0 )
return 0 ;
}
}
return (p % 2 == 0 )? - 1 : 1 ;
}
// Driver code
public static void main (String[] args)
{
int N = 17 ;
System.out.println( "Mobius Functions"
+ " M(N) at N = " + N + " is: "
+ mobius(N));
System.out.println ( "Mobius Functions"
+ "M(N) at N = " + 25 + " is: "
+ mobius( 25 ));
System.out.println( "Mobius Functions"
+ "M(N) at N = " + 6 + " is: "
+ mobius( 6 ));
}
} // This code is contributed by anuj_67. |
// C# program to print all prime factors using System;
class GFG {
// Returns value of mobius()
static int mobius( int n)
{
int p = 0;
// Handling 2 separately
if (n % 2 == 0)
{
n = n / 2;
p++;
// If 2^2 also divides N
if (n % 2 == 0)
return 0;
}
// Check for all other prime factors
for ( int i = 3; i <= Math.Sqrt(n);
i = i+2)
{
// If i divides n
if (n % i == 0)
{
n = n / i;
p++;
// If i^2 also divides N
if (n % i == 0)
return 0;
}
}
return (p % 2 == 0)? -1 : 1;
}
// Driver Code
public static void Main ()
{
int N = 17;
Console.WriteLine( "Mobius Functions"
+ " M(N) at N = " + N + " is: "
+ mobius(N));
Console.WriteLine( "Mobius Functions"
+ "M(N) at N = " + 25 + " is: "
+ mobius(25));
Console.WriteLine( "Mobius Functions"
+ "M(N) at N = " + 6 + " is: "
+ mobius(6));
}
} // This code is contributed by anuj_67. |
<?php // PHP Program to print // all prime factors // Returns value of mobius() function mobius( $n )
{ $p = 0;
// Handling 2 separately
if ( $n % 2 == 0)
{
$n = $n / 2;
$p ++;
// If 2^2 also divides N
if ( $n % 2 == 0)
return 0;
}
// Check for all
// other prime factors
for ( $i = 3; $i <= sqrt( $n ); $i = $i + 2)
{
// If i divides n
if ( $n % $i == 0)
{
$n = $n / $i ;
$p ++;
// If i^2 also divides N
if ( $n % $i == 0)
return 0;
}
}
return ( $p % 2 == 0)? -1 : 1;
} // Driver code
$N = 17;
echo "Mobius Functions M(N) at N = " , $N , " is: "
, mobius( $N ), "\n" ;
echo "Mobius Functions M(N) at N = " , 25 , " is: "
, mobius(25), "\n" ;
echo "Mobius Functions M(N) at N = " , 6 , " is: "
, mobius(6) ;
// This code is contributed by anuj_67. ?> |
<script> // JavaScript program to print all prime factors
// Returns value of mobius()
function mobius(n)
{
let p = 0;
// Handling 2 separately
if (n % 2 == 0)
{
n = parseInt(n / 2, 10);
p++;
// If 2^2 also divides N
if (n % 2 == 0)
return 0;
}
// Check for all other prime factors
for (let i = 3; i <= Math.sqrt(n); i = i+2)
{
// If i divides n
if (n % i == 0)
{
n = parseInt(n / i, 10);
p++;
// If i^2 also divides N
if (n % i == 0)
return 0;
}
}
return (p % 2 == 0)? -1 : 1;
}
let N = 17;
document.write( "Mobius Functions"
+ " M(N) at N = " + N + " is: "
+ mobius(N) + "</br>" );
document.write( "Mobius Functions"
+ "M(N) at N = " + 25 + " is: "
+ mobius(25) + "</br>" );
document.write( "Mobius Functions"
+ "M(N) at N = " + 6 + " is: "
+ mobius(6));
</script> |
Output:
Mobius Functions M(N) at N = 17 is: -1 Mobius Functions M(N) at N = 25 is: 0 Mobius Functions M(N) at N = 6 is: 1
Time Complexity: O(?n)
Auxiliary Space: O(1)
Please suggest if someone has a better solution which is more efficient in terms of space and time.
References
1) http://mathworld.wolfram.com/MobiusFunction.html
2) https://en.wikipedia.org/wiki/M%C3%B6bius_function
3) https://en.wikipedia.org/wiki/Completely_multiplicative_function