Suppose we have a number n; we have to check whether n is an Achilles number or not. As we know a number is Achilles number when a number that is powerful (A number N is called Powerful Number when for every prime factor p of it, p^2 also divides it) but not a perfect power. Some of the examples of Achilles numbers are: 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125.
So, if the input is like 108, then the output will be True, as 6 and 36 both divide it and it is not perfect square.
To solve this, we will follow these steps −
- Define a function check_powerful() . This will take n
- while n mod 2 is same, do
- p := 0
- while n mod 2 is same as 0, do
- n := n / 2
- p := p + 1
- if p is same as 1, then
- return False
- p := integer of (square root of n) + 1
- for factor in range 3 to p, increase by 2, do
- p := 0
- while n mod factor is same as 0, do
- n := n / factor
- p := p + 1
- if p is same as 1, then
- return False
- return true when (n is same as 1)
- Define a function check_power() . This will take a
- if a is same as 1, then
- return True
- p := integer of (square root of n) + 1
- for i in range 2 to a, increase by 1, do
- val := log(a) / log(i) [all base e]
- if (val - integer part of (val)) < 0.00000001, then
- return True
- return False
- From the main method, do the following −
- if check_powerful(n) is same as True and check_power(n) is same as False, then
- return True
- otherwise,
- return False
Example
Let us see the following implementation to get better understanding −
from math import sqrt, log def check_powerful(n): while (n % 2 == 0): p = 0 while (n % 2 == 0): n /= 2 p += 1 if (p == 1): return False p = int(sqrt(n)) + 1 for factor in range(3, p, 2): p = 0 while (n % factor == 0): n = n / factor p += 1 if (p == 1): return False return (n == 1) def check_power(a): if (a == 1): return True p = int(sqrt(a)) + 1 for i in range(2, a, 1): val = log(a) / log(i) if ((val - int(val)) < 0.00000001): return True return False def isAchilles(n): if (check_powerful(n) == True and check_power(n) == False): return True else: return False n = 108 print(isAchilles(n))
Input
108
Output
True