Write a java program for a given number n, the task is to find the odd factor sum.
Examples:
Input: n = 30
Output: 24
Explanation: Odd dividers sum 1 + 3 + 5 + 15 = 24Input: 18
Output: 13
Explanation: Odd dividers sum 1 + 3 + 9 = 13
Java program for Find sum of odd factors of a number using Prime factorization:
To find sum of odd factors, we simply need to ignore even factors and their powers. For example, consider n = 18. It can be written as 2132 and sum of all factors is (1)*(1 + 2)*(1 + 3 + 32). Sum of odd factors (1)*(1+3+32) = 13.
To remove all even factors, we repeatedly divide n while it is divisible by 2. After this step, we only get odd factors. Note that 2 is the only even prime.
Let p1, p2, … pk be prime factors of n. Let a1, a2, .. ak be highest powers of p1, p2, .. pk respectively that divide n, i.e., we can write n as n = (p1a1)*(p2a2)* … (pkak).
Sum of divisors = (1 + p1 + p12 ... p1a1) * (1 + p2 + p22 ... p2a2) * ............................................. (1 + pk + pk2 ... pkak)
Below is the Implementation of the above approach:
// Formula based Java program // to find sum of all divisors // of n. import java.io.*;
import java.math.*;
class GFG {
// Returns sum of all
// factors of n.
static int sumofoddFactors( int n)
{
// Traversing through
// all prime factors.
int res = 1 ;
// ignore even factors by
// removing all powers
// of 2
while (n % 2 == 0 )
n = n / 2 ;
for ( int i = 3 ; i <= Math.sqrt(n); i++) {
// While i divides n, print i
// and divide n
int count = 0 , curr_sum = 1 ;
int curr_term = 1 ;
while (n % i == 0 ) {
count++;
n = n / i;
curr_term *= i;
curr_sum += curr_term;
}
res *= curr_sum;
}
// This condition is to handle
// the case when n is a
// prime number.
if (n >= 2 )
res *= ( 1 + n);
return res;
}
// Driver code
public static void main(String args[])
throws IOException
{
int n = 30 ;
System.out.println(sumofoddFactors(n));
}
} /* This code is contributed by Nikita Tiwari.*/ |
Output:
24
Time Complexity: O(sqrt(n))
Auxiliary Space: O(1)
Please refer complete article on Find sum of odd factors of a number for more details!