Given a Number
Trojan Number is a number that is a strong number but not a perfect power. A number N is known as a strong number if, for every prime divisor or factor p of N, p2 is also a divisor. In other words, every prime factor appears at least twice.
All Trojan numbers are strong. However, not all strong numbers are Trojan numbers: only those that cannot be represented as mk, where m and k are positive integers greater than 1.
Examples:
Input : N = 108 Output : YES Input : N = 8 Output : NO
The idea is to store the count of each prime factor and check if the count is greater than 2 then it will be a Strong Number.
This part can easily be calculated by prime factorization through sieve.
The next step is to check if the given number cannot be expressed as xy. To check whether a number is perfect power or not refer to this article.
Below is the implementation of above problem:
// CPP program to check if a number is // Trojan Number or not #include <bits/stdc++.h> using namespace std;
// Function to check if a number // can be expressed as x^y bool isPerfectPower( int n)
{ if (n == 1)
return true ;
// Try all numbers from 2 to sqrt(n) as base
for ( int x = 2; x <= sqrt (n); x++) {
int y = 2;
int p = pow (x, y);
// Keep increasing y while power 'p'
// is smaller than n.
while (p <= n && p > 0) {
if (p == n)
return true ;
y++;
p = pow (x, y);
}
}
return false ;
} // Function to check if a number is Strong bool isStrongNumber( int n)
{ unordered_map< int , int > count;
while (n % 2 == 0) {
n = n / 2;
count[2]++;
}
// count the number for each prime factor
for ( int i = 3; i <= sqrt (n); i += 2) {
while (n % i == 0) {
n = n / i;
count[i]++;
}
}
if (n > 2)
count[n]++;
int flag = 0;
for ( auto b : count) {
// minimum number of prime divisors
// should be 2
if (b.second == 1) {
flag = 1;
break ;
}
}
if (flag == 1)
return false ;
else
return true ;
} // Function to check if a number // is Trojan Number bool isTrojan( int n)
{ if (!isPerfectPower(n) && isStrongNumber(n))
return true ;
else
return false ;
} // Driver Code int main()
{ int n = 108;
if (isTrojan(n))
cout << "YES" ;
else
cout << "NO" ;
return 0;
} |
// Java program to check if a number is // Trojan Number or not import java.util.*;
class GFG
{ // Function to check if a number
// can be expressed as x^y
static boolean isPerfectPower( int n)
{
if (n == 1 )
{
return true ;
}
// Try all numbers from 2 to sqrt(n) as base
for ( int x = 2 ; x <= Math.sqrt(n); x++)
{
int y = 2 ;
int p = ( int ) Math.pow(x, y);
// Keep increasing y while power 'p'
// is smaller than n.
while (p <= n && p > 0 )
{
if (p == n)
{
return true ;
}
y++;
p = ( int ) Math.pow(x, y);
}
}
return false ;
}
// Function to check if a number is Strong
static boolean isStrongNumber( int n)
{
HashMap<Integer,
Integer> count = new HashMap<Integer,
Integer>();
while (n % 2 == 0 )
{
n = n / 2 ;
if (count.containsKey( 2 ))
{
count.put( 2 , count.get( 2 ) + 1 );
}
else
{
count.put( 2 , 1 );
}
}
// count the number for each prime factor
for ( int i = 3 ; i <= Math.sqrt(n); i += 2 )
{
while (n % i == 0 )
{
n = n / i;
if (count.containsKey(i))
{
count.put(i, count.get(i) + 1 );
}
else
{
count.put(i, 1 );
}
}
}
if (n > 2 )
{
if (count.containsKey(n))
{
count.put(n, count.get(n) + 1 );
}
else
{
count.put(n, 1 );
}
}
int flag = 0 ;
for (Map.Entry<Integer,
Integer> b : count.entrySet())
{
// minimum number of prime divisors
// should be 2
if (b.getValue() == 1 )
{
flag = 1 ;
break ;
}
}
if (flag == 1 )
{
return false ;
}
else
{
return true ;
}
}
// Function to check if a number
// is Trojan Number
static boolean isTrojan( int n)
{
if (!isPerfectPower(n) && isStrongNumber(n))
{
return true ;
}
else
{
return false ;
}
}
// Driver Code
public static void main(String[] args)
{
int n = 108 ;
if (isTrojan(n))
{
System.out.println( "Yes" );
}
else
{
System.out.println( "No" );
}
}
} // This code is contributed by PrinciRaj1992 |
# Python 3 program to check if a number # is Trojan Number or not from math import sqrt, pow
# Function to check if a number # can be expressed as x^y def isPerfectPower(n):
if n = = 1 :
return True
# Try all numbers from 2 to
# sqrt(n) as base
for x in range ( 2 , int (sqrt(n)) + 1 ):
y = 2
p = pow (x, y)
# Keep increasing y while power
# 'p' is smaller than n.
while p < = n and p > 0 :
if p = = n:
return True
y + = 1
p = pow (x, y)
return False
# Function to check if a number # is Strong def isStrongNumber(n):
count = {i: 0 for i in range (n)}
while n % 2 = = 0 :
n = n / / 2
count[ 2 ] + = 1
# count the number for each
# prime factor
for i in range ( 3 , int (sqrt(n)) + 1 , 2 ):
while n % i = = 0 :
n = n / / i
count[i] + = 1
if n > 2 :
count[n] + = 1
flag = 0
for key,value in count.items():
# minimum number of prime
# divisors should be 2
if value = = 1 :
flag = 1
break
if flag = = 1 :
return False
return True
# Function to check if a number # is Trojan Number def isTrojan(n):
return isPerfectPower(n) = = False and isStrongNumber(n)
# Driver Code if __name__ = = '__main__' :
n = 108
if (isTrojan(n)):
print ( "YES" )
else :
print ( "NO" )
# This code is contributed by # Surendra_Gangwar |
// C# program to check if a number is // Trojan Number or not using System;
using System.Collections.Generic;
class GFG
{ // Function to check if a number
// can be expressed as x^y
static bool isPerfectPower( int n)
{
if (n == 1)
{
return true ;
}
// Try all numbers from 2 to sqrt(n) as base
for ( int x = 2; x <= Math.Sqrt(n); x++)
{
int y = 2;
int p = ( int ) Math.Pow(x, y);
// Keep increasing y while power 'p'
// is smaller than n.
while (p <= n && p > 0)
{
if (p == n)
{
return true ;
}
y++;
p = ( int ) Math.Pow(x, y);
}
}
return false ;
}
// Function to check if a number is Strong
static bool isStrongNumber( int n)
{
Dictionary< int ,
int > count = new Dictionary< int ,
int >();
while (n % 2 == 0)
{
n = n / 2;
if (count.ContainsKey(2))
{
count[2] = count[2] + 1;
}
else
{
count.Add(2, 1);
}
}
// count the number for each prime factor
for ( int i = 3; i <= Math.Sqrt(n); i += 2)
{
while (n % i == 0)
{
n = n / i;
if (count.ContainsKey(i))
{
count[i] = count[i] + 1;
}
else
{
count.Add(i, 1);
}
}
}
if (n > 2)
{
if (count.ContainsKey(n))
{
count[n] = count[n] + 1;
}
else
{
count.Add(n, 1);
}
}
int flag = 0;
foreach (KeyValuePair< int , int > b in count)
{
// minimum number of prime divisors
// should be 2
if (b.Value == 1)
{
flag = 1;
break ;
}
}
if (flag == 1)
{
return false ;
}
else
{
return true ;
}
}
// Function to check if a number
// is Trojan Number
static bool isTrojan( int n)
{
if (!isPerfectPower(n) &&
isStrongNumber(n))
{
return true ;
}
else
{
return false ;
}
}
// Driver Code
public static void Main(String[] args)
{
int n = 108;
if (isTrojan(n))
{
Console.WriteLine( "Yes" );
}
else
{
Console.WriteLine( "No" );
}
}
} // This code is contributed by Princi Singh |
<script> // javascript program to check if a number is // Trojan Number or not // Function to check if a number
// can be expressed as x^y
function isPerfectPower(n) {
if (n == 1) {
return true ;
}
// Try all numbers from 2 to sqrt(n) as base
for ( var x = 2; x <= Math.sqrt(n); x++) {
var y = 2;
var p = parseInt( Math.pow(x, y));
// Keep increasing y while power 'p'
// is smaller than n.
while (p <= n && p > 0) {
if (p == n) {
return true ;
}
y++;
p = parseInt( Math.pow(x, y));
}
}
return false ;
}
// Function to check if a number is Strong
function isStrongNumber(n) {
var count = new Map();
while (n % 2 == 0) {
n = n / 2;
if (count.has(2)) {
count.set(2, count.get(2) + 1);
} else {
count.set(2, 1);
}
}
// count the number for each prime factor
for ( var i = 3; i <= Math.sqrt(n); i += 2) {
while (n % i == 0) {
n = n / i;
if (count.has(i)) {
count.set(i, count.get(i) + 1);
} else {
count.set(i, 1);
}
}
}
if (n > 2) {
if (count.has(n)) {
count.set(n, count.get(n) + 1);
} else {
count.set(n, 1);
}
}
var flag = 0;
const iterator = count[Symbol.iterator]();
let itr = iterator.next()
for (let i = 0; i < count.size; i++) {
console.log(itr.value, itr.done)
if (itr.value == 1) {
flag = 1;
break ;
}
itr = iterator.next()
}
if (flag == 1) {
return false ;
} else {
return true ;
}
}
// Function to check if a number
// is Trojan Number
function isTrojan(n) {
if (!isPerfectPower(n) && isStrongNumber(n)) {
return true ;
} else {
return false ;
}
}
// Driver Code
var n = 108;
if (isTrojan(n)) {
document.write( "Yes" );
} else {
document.write( "No" );
}
// This code contributed by gauravrajput1 </script> |
Output:
YES
Auxiliary Space: O(n)