number of factors of n that are less than square root of n

Question

i want to find number of factors of a number say 900 that are less than its square root. eg: there are 27 factors of 900 and i want to find number of factors smaller than root of 900 ie, 30 which are 1,2,3,4,5,6,9,10,12,15,18,20,25.

i currently have this program that finds the number of factors by calculating the number of prime factors. eg:prime factors of 140 are:2^2*5*7. So the number of factors are:(2+1)(1+1)(1+1) [multiplication of powers of prime factors]

import java.io.*;
import java.util.*;
class Solution
{
// Program to print all prime factors
static void primeFactors(int n)
{

    TreeMap tm=new TreeMap();
    int times=0;
    // Print the number of 2s that divide n
    while (n%2 == 0)
    {
        System.out.println("2");
        if(!tm.containsKey(2))
        {
            tm.put(2,1);
        }
        else
        {
            times=(int)tm.get(2);
            tm.put(2,times+1);
        }
        n = n/2;
    }

    // n must be odd at this point.  So we can skip one element (Note i = i +2)
    for (int i = 3; i <= Math.sqrt(n); i = i+2)
    {
        // While i divides n, print i and divide n
        while (n%i == 0)
        {
            System.out.println(i);
            if(!tm.containsKey(i))
            {
                tm.put(i,1);
            }
            else
            {
            times=(int)tm.get(i);
            tm.put(i,times+1);
            }
            n = n/i;
        }
    }

    // This condition is to handle the case whien n is a prime number
    // greater than 2
    if (n > 2)
    {
        System.out.println(n);
        if(!tm.containsKey(n))
        {
            tm.put(n,1);
        }
        else
        {
        times=(int)tm.get(n);
        tm.put(n,times+1);
        }
    }

    /////////////////////////////////////////////////////////////////////////////
    Set set = tm.entrySet();
    System.out.println(tm);
    Iterator num = set.iterator();
    int key=0;
    int sum=1;
    while (num.hasNext())
    {
        Map.Entry number =(Map.Entry)num.next();
        sum=sum*((int)number.getValue()+1);
    }
    System.out.println(sum);
}

public static void main(String args[])
{
    Scanner sc=new Scanner(System.in);
    int n=sc.nextInt();
    primeFactors(n);
}
}

here i am getting number of factors eg:27 factors for 900 but i want to find number of factors that are less than 30. Thanks for help.

Answer 1

If you have the number of factors of n, simply integer divide by 2 to get the number of factors less than the square root. This works because each factor d of n less than sqrt(n) corresponds to a factor greater than sqrt(n) (namely n/d), so the number of such factors will be half the total (unless n is a perfect square, in which case sqrt(n) is an extra factor). However, integer division by 2 takes care of that corner case. Indeed, 27/2 = 13 as desired.