Given three integers a, b, n .Your task is to print number of numbers between a and b including them also which have n-divisors. A number is called n-divisor if it has total n divisors including 1 and itself.
Examples:
Input : a = 1, b = 7, n = 2 Output : 4 There are four numbers with 2 divisors in range [1, 7]. The numbers are 2, 3, 5, and 7.
Naive Approach:
The naive approach is to check all the numbers between a and b how many of them are n-divisor number for doing this find out the number of each divisors for each number . If it is equal to n then it is a n-divisor number
Efficient Approach:
Any number can be written in the form of its prime factorization let the number be x and p1, p2..pm are the prime numbers which divide x so x = p1e1 * p2e2....pmem where e1, e2...em are the exponents of prime numbers p1, p2....pm. So the number of divisors of x will be (e1+1)*(e2+1)...*(em+1).
Now the second observation is for prime numbers greater than sqrt(x) their exponent cannot exceed 1. Let's prove this by contradiction suppose there is a prime number P greater than sqrt(x) and its exponent E in prime factorization of x is greater than one (E >= 2) so P^E sqrt(x) so P^E > (sqrt(x))E and E >= 2 so PE will always be greater than x
Third observation is that number of prime numbers greater than sqrt(x) in the prime factorization of x will always be less than equal to 1. This can also be proved similarly by contradiction as above.
Now to solve this problem
Step 1: Apply sieve of eratosthenes and calculate prime numbers upto sqrt(b).
Step 2: Traverse through each number from a to b and calculate exponents of each prime number in that number by repeatedly dividing that number by prime number and use the formula numberofdivisors(x) = (e1+1)*(e2+1)....(em+1).
Step 3: If after dividing by all the prime numbers less than equal to square root of that number if number > 1 this means there is a prime number greater than its square root which divides and its exponent will always be one as proved above.
Try It Yourself
// C++ program to count numbers with n divisors
#include<bits/stdc++.h>
using namespace std;
// applying sieve of eratosthenes
void sieve(bool primes[], int x)
{
primes[1] = false;
// if a number is prime mark all its multiples
// as non prime
for (int i=2; i*i <= x; i++)
{
if (primes[i] == true)
{
for (int j=2; j*j <= x; j++)
primes[i*j] = false;
}
}
}
// function that returns numbers of number that have
// n divisors in range from a to b. x is sqrt(b) + 1.
int nDivisors(bool primes[], int x, int a, int b, int n)
{
// result holds number of numbers having n divisors
int result = 0;
// vector to hold all the prime numbers between 1
// ans sqrt(b)
vector <int> v;
for (int i = 2; i <= x; i++)
if (primes[i] == true)
v.push_back (i);
// Traversing all numbers in given range
for (int i=a; i<=b; i++)
{
// initialising temp as i
int temp = i;
// total holds the number of divisors of i
int total = 1;
int j = 0;
// we need to use that prime numbers that
// are less than equal to sqrt(temp)
for (int k = v[j]; k*k <= temp; k = v[++j])
{
// holds the exponent of k in prime
// factorization of i
int count = 0;
// repeatedly divide temp by k till it is
// divisible and accordingly increase count
while (temp%k == 0)
{
count++;
temp = temp/k;
}
// using the formula no.of divisors =
// (e1+1)*(e2+1)....
total = total*(count+1);
}
// if temp is not equal to 1 then there is
// prime number in prime factorization of i
// greater than sqrt(i)
if (temp != 1)
total = total*2;
// if i is a ndvisor number increase result
if (total == n)
result++;
}
return result;
}
// Returns count of numbers in [a..b] having
// n divisors.
int countNDivisors(int a, int b, int n)
{
int x = sqrt(b) + 1;
// primes[i] = true if i is a prime number
bool primes[x];
// initialising each number as prime
memset(primes, true, sizeof(primes));
sieve(primes, x);
return nDivisors(primes, x, a, b, n);
}
// driver code
int main()
{
int a = 1, b = 7, n = 2;
cout << countNDivisors(a, b, n);
return 0;
}
// Java program to count numbers with n divisors
import java.util.*;
class GFG{
// applying sieve of eratosthenes
static void sieve(boolean[] primes, int x)
{
primes[1] = true;
// if a number is prime mark all its multiples
// as non prime
for (int i=2; i*i <= x; i++)
{
if (primes[i] == false)
{
for (int j=2; j*i <= x; j++)
primes[i*j] = true;
}
}
}
// function that returns numbers of number that have
// n divisors in range from a to b. x is sqrt(b) + 1.
static int nDivisors(boolean[] primes, int x, int a, int b, int n)
{
// result holds number of numbers having n divisors
int result = 0;
// vector to hold all the prime numbers between 1
// ans sqrt(b)
ArrayList<Integer> v=new ArrayList<Integer>();
for (int i = 2; i <= x; i++)
if (primes[i] == false)
v.add(i);
// Traversing all numbers in given range
for (int i=a; i<=b; i++)
{
// initialising temp as i
int temp = i;
// total holds the number of divisors of i
int total = 1;
int j = 0;
// we need to use that prime numbers that
// are less than equal to sqrt(temp)
for (int k = v.get(j); k*k <= temp; k = v.get(++j))
{
// holds the exponent of k in prime
// factorization of i
int count = 0;
// repeatedly divide temp by k till it is
// divisible and accordingly increase count
while (temp%k == 0)
{
count++;
temp = temp/k;
}
// using the formula no.of divisors =
// (e1+1)*(e2+1)....
total = total*(count+1);
}
// if temp is not equal to 1 then there is
// prime number in prime factorization of i
// greater than sqrt(i)
if (temp != 1)
total = total*2;
// if i is a ndvisor number increase result
if (total == n)
result++;
}
return result;
}
// Returns count of numbers in [a..b] having
// n divisors.
static int countNDivisors(int a, int b, int n)
{
int x = (int)Math.sqrt(b) + 1;
// primes[i] = true if i is a prime number
boolean[] primes=new boolean[x+1];
// initialising each number as prime
sieve(primes, x);
return nDivisors(primes, x, a, b, n);
}
// driver code
public static void main(String[] args)
{
int a = 1, b = 7, n = 2;
System.out.println(countNDivisors(a, b, n));
}
}
// This code is contributed by mits
# Python3 program to count numbers
# with n divisors
import math;
# applying sieve of eratosthenes
def sieve(primes, x):
primes[1] = False;
# if a number is prime mark all
# its multiples as non prime
i = 2;
while (i * i <= x):
if (primes[i] == True):
j = 2;
while (j * i <= x):
primes[i * j] = False;
j += 1;
i += 1;
# function that returns numbers of number
# that have n divisors in range from a to b.
# x is sqrt(b) + 1.
def nDivisors(primes, x, a, b, n):
# result holds number of numbers
# having n divisors
result = 0;
# vector to hold all the prime
# numbers between 1 and sqrt(b)
v = [];
for i in range(2, x + 1):
if (primes[i]):
v.append(i);
# Traversing all numbers in given range
for i in range(a, b + 1):
# initialising temp as i
temp = i;
# total holds the number of
# divisors of i
total = 1;
j = 0;
# we need to use that prime numbers that
# are less than equal to sqrt(temp)
k = v[j];
while (k * k <= temp):
# holds the exponent of k in prime
# factorization of i
count = 0;
# repeatedly divide temp by k till it is
# divisible and accordingly increase count
while (temp % k == 0):
count += 1;
temp = int(temp / k);
# using the formula no.of divisors =
# (e1+1)*(e2+1)....
total = total * (count + 1);
j += 1;
k = v[j];
# if temp is not equal to 1 then there is
# prime number in prime factorization of i
# greater than sqrt(i)
if (temp != 1):
total = total * 2;
# if i is a ndivisor number
# increase result
if (total == n):
result += 1;
return result;
# Returns count of numbers in [a..b]
# having n divisors.
def countNDivisors(a, b, n):
x = int(math.sqrt(b) + 1);
# primes[i] = true if i is a prime number
# initialising each number as prime
primes = [True] * (x + 1);
sieve(primes, x);
return nDivisors(primes, x, a, b, n);
# Driver code
a = 1;
b = 7;
n = 2;
print(countNDivisors(a, b, n));
# This code is contributed by mits
// C# program to count numbers with n divisors
using System.Collections;
using System;
class GFG{
// applying sieve of eratosthenes
static void sieve(bool[] primes, int x)
{
primes[1] = true;
// if a number is prime mark all its multiples
// as non prime
for (int i=2; i*i <= x; i++)
{
if (primes[i] == false)
{
for (int j=2; j*i <= x; j++)
primes[i*j] = true;
}
}
}
// function that returns numbers of number that have
// n divisors in range from a to b. x is sqrt(b) + 1.
static int nDivisors(bool[] primes, int x, int a, int b, int n)
{
// result holds number of numbers having n divisors
int result = 0;
// vector to hold all the prime numbers between 1
// ans sqrt(b)
ArrayList v=new ArrayList();
for (int i = 2; i <= x; i++)
if (primes[i] == false)
v.Add(i);
// Traversing all numbers in given range
for (int i=a; i<=b; i++)
{
// initialising temp as i
int temp = i;
// total holds the number of divisors of i
int total = 1;
int j = 0;
// we need to use that prime numbers that
// are less than equal to sqrt(temp)
for (int k = (int)v[j]; k*k <= temp; k = (int)v[++j])
{
// holds the exponent of k in prime
// factorization of i
int count = 0;
// repeatedly divide temp by k till it is
// divisible and accordingly increase count
while (temp%k == 0)
{
count++;
temp = temp/k;
}
// using the formula no.of divisors =
// (e1+1)*(e2+1)....
total = total*(count+1);
}
// if temp is not equal to 1 then there is
// prime number in prime factorization of i
// greater than sqrt(i)
if (temp != 1)
total = total*2;
// if i is a ndivisor number increase result
if (total == n)
result++;
}
return result;
}
// Returns count of numbers in [a..b] having
// n divisors.
static int countNDivisors(int a, int b, int n)
{
int x = (int)Math.Sqrt(b) + 1;
// primes[i] = true if i is a prime number
bool[] primes=new bool[x+1];
// initialising each number as prime
sieve(primes, x);
return nDivisors(primes, x, a, b, n);
}
// driver code
public static void Main()
{
int a = 1, b = 7, n = 2;
Console.WriteLine(countNDivisors(a, b, n));
}
}
// This code is contributed by mits
<?php
// PHP program to count numbers with n divisors
// applying sieve of eratosthenes
function sieve(&$primes, $x)
{
$primes[1] = false;
// if a number is prime mark all
// its multiples as non prime
for ($i = 2; $i * $i <= $x; $i++)
{
if ($primes[$i] == true)
{
for ($j = 2; $j * $i <= $x; $j++)
$primes[$i * $j] = false;
}
}
}
// function that returns numbers of number
// that have n divisors in range from a to
// b. x is sqrt(b) + 1.
function nDivisors($primes, $x, $a, $b, $n)
{
// result holds number of numbers
// having n divisors
$result = 0;
// vector to hold all the prime numbers
// between 1 ans sqrt(b)
$v = array();
for ($i = 2; $i <= $x; $i++)
if ($primes[$i] == true)
array_push($v, $i);
// Traversing all numbers in given range
for ($i = $a; $i <= $b; $i++)
{
// initialising temp as i
$temp = $i;
// total holds the number of
// divisors of i
$total = 1;
$j = 0;
// we need to use that prime numbers that
// are less than equal to sqrt(temp)
for ($k = $v[$j];
$k * $k <= $temp; $k = $v[++$j])
{
// holds the exponent of k in
// prime factorization of i
$count = 0;
// repeatedly divide temp by k till
// it is divisible and accordingly
// increase count
while ($temp % $k == 0)
{
$count++;
$temp = (int)($temp / $k);
}
// using the formula no.of divisors =
// (e1+1)*(e2+1)....
$total = $total * ($count + 1);
}
// if temp is not equal to 1 then there is
// prime number in prime factorization of i
// greater than sqrt(i)
if ($temp != 1)
$total = $total * 2;
// if i is a n divisor number increase result
if ($total == $n)
$result++;
}
return $result;
}
// Returns count of numbers in [a..b]
// having n divisors.
function countNDivisors($a, $b, $n)
{
$x = (int)(sqrt($b) + 1);
// primes[i] = true if i is a prime number
// initialising each number as prime
$primes = array_fill(0, $x + 1, true);
sieve($primes, $x);
return nDivisors($primes, $x, $a, $b, $n);
}
// Driver code
$a = 1;
$b = 7;
$n = 2;
print(countNDivisors($a, $b, $n));
// This code is contributed by mits
?>
<script>
// Javascript program to count numbers with n divisors
// applying sieve of eratosthenes
function sieve(primes, x)
{
primes[1] = true;
// if a number is prime mark all its multiples
// as non prime
for (var i=2; i*i <= x; i++)
{
if (primes[i] == false)
{
for (var j=2; j*i <= x; j++)
primes[i*j] = true;
}
}
}
// function that returns numbers of number that have
// n divisors in range from a to b. x is sqrt(b) + 1.
function nDivisors(primes, x, a, b, n)
{
// result holds number of numbers having n divisors
var result = 0;
// vector to hold all the prime numbers between 1
// ans sqrt(b)
var v = [];
for (var i = 2; i <= x; i++)
if (primes[i] == false)
v.push(i);
// Traversing all numbers in given range
for (var i=a; i<=b; i++)
{
// initialising temp as i
var temp = i;
// total holds the number of divisors of i
var total = 1;
var j = 0;
// we need to use that prime numbers that
// are less than equal to sqrt(temp)
for (var k = v[j]; k*k <= temp; k = v[++j])
{
// holds the exponent of k in prime
// factorization of i
var count = 0;
// repeatedly divide temp by k till it is
// divisible and accordingly increase count
while (temp%k == 0)
{
count++;
temp = parseInt(temp/k);
}
// using the formula no.of divisors =
// (e1+1)*(e2+1)....
total = total*(count+1);
}
// if temp is not equal to 1 then there is
// prime number in prime factorization of i
// greater than sqrt(i)
if (temp != 1)
total = total*2;
// if i is a ndivisor number increase result
if (total == n)
result++;
}
return result;
}
// Returns count of numbers in [a..b] having
// n divisors.
function countNDivisors(a, b, n)
{
var x = parseInt(Math.sqrt(b)) + 1;
// primes[i] = true if i is a prime number
var primes = Array(x+1).fill(false);
// initialising each number as prime
sieve(primes, x);
return nDivisors(primes, x, a, b, n);
}
// driver code
var a = 1, b = 7, n = 2;
document.write(countNDivisors(a, b, n));
// This code is contributed by rutvik_56.
</script>
Output:
4
Time Complexity: O(n)
Auxiliary Space: O(n)