Achilles Number in Java

In this section, we will learn what is an Achilles number and also create Java programs to check if the given number is an Achilles number or not. The Achilles number program frequently asked in Java coding tests and academics.

Achilles Number

A number that is a powerful number but not a perfect power is known as Achilles number. It is an integer sequence A052486 in OEIS.

Powerful Numbers: It is a positive integer such that for every prime factor p of N, p² is also divided it. For example, the number 36 is a powerful number because it is divisible by both 3 and square of 3, i.e. 9. Some other powerful numbers are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, etc.
Perfect Power: It is also a positive integer such that it can be expressed as a^b. Where a and b are integers, and b is greater than 1 and a is greater than 0. For example, 64 can be expressed as 8².

Let's understand it mathematically through an example.

Achilles Number Example

Let's check the number 108 is an Achilles number or not.

The factors of 108 = 2 x 2 x 3 x 3 x 3.

Prime factors are 2 and 3.

Square of prime factors are:
Both 2 and 2² = 4 divides 108.
Both 3 and 3² = 9 divides 108.
But, the number 108 cannot be represented as a^b.
Hence, 108 is an Achilles number.

Let's take another number 784 and check it is an Achilles number or not.

The factors of 784 = 2 x 2 x 2 x 2 x 7 x 7.

Prime factors are 2 and 7.

Square of prime factors are:
Both 2 and 2² = 4 divides 784.
Both 7 and 7² = 49 divides 784.
But, the number 784 can be represented as 28².
Hence, 784 is not an Achilles number.

Similarly, we can check other numbers. First few Achilles numbers are 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000, etc.

Steps to Find Achilles Number

Read or initialize a number N.
Check if the given number N is powerful or not.
Check if the given number N is perfect power or square or not.
If N is powerful but not perfect, the number N is an Achilles number, else not.

Let's implement the above steps in a Java program.

Achilles Number Java Program

AchillesNumberExample.java

import java.util.Scanner;
public class AchillesNumberExample 
{
//function that checks whether the given number is powerful or not
static boolean isPowerful(int n)
{
//repeatedly divides the given number by 2
while (n % 2 == 0) 
{
int power = 0;
//executes until the condition returns false
while (n % 2 == 0) 
{
n = n/ 2;
//increments power by 1
power++;
}
//if only 2 to the power 1 (not higher powers) divides the number 
if (power == 1)
//returns false if power equals to 1
return false;
}
// if n is not a power of 2 then this loop
// will execute repeat above process
for (int factor = 3; factor <= Math.sqrt(n); factor += 2) 
{
int power = 0;
//determines the highest power of factor that divides the number
//the loop executes until the condition returns false
while (n % factor == 0) 
{
//divides the number by factor
n = n / factor;
power++;
}
//if the power of factor is 1 (not higher powers) and it divides the number n then returns false
if (power == 1)
return false;
}
//the value of n should be 1
//returns false if n is not equal to 1 because prime numbers are not powerful 
return (n == 1);
}
//function that checks if the number is a perfect power or not
static boolean isPower(int a)
{
if (a == 1)
return true;
for (int i = 2; i * i <= a; i++) 
{
double val = Math.log(a) / Math.log(i);
if ((val - (int)val) < 0.00000001)
return true;
}
return false;
}
//function that checks if the given number is Achilles Number or not
static boolean isAchilles(int n)
{
//checks if the number is powerful and not a perfect power 
if (isPowerful(n) && !isPower(n))
//returns true if both conditions satisfy 
return true;
else
//returns false if any of the condition returns false
return false;
}
//driver code
public static void main(String args[])
{
Scanner sc = new Scanner(System.in);
System.out.print("Enter the number: ");
//reading an integer from the user
int n=sc.nextInt();
//calling the user-defined function that checks if the number if Achilles or not
if (isAchilles(n))
System.out.println("The given number is an Achilles number.");
else
System.out.println("The given number is not an Achilles number.");
sc.close();
}
}

Output 1: