In the realm of mathematical theory, a "super-perfect number" is an extension of the concept of perfect numbers. A superperfect number is defined as a positive integer, denoted as n, where 2n encompasses all divisors of n, including n itself.
A superperfect number is a number that meets the criteria outlined by the following formula:
sig(sig(n)) = 2n
Here, sig(n) is referred to as the divisor summatory function, which calculates the total sum of a number's divisors.
Example:
Let's verify if the value N qualifies as a superperfect number or not:
Input:
N = 16
Output:
yes
Explanation:
The function sig(16) calculates the sum of all divisors of 16, which are 1, 2, 4, 8, and 16, resulting in a total of 31.
sig(31)=1+31=32(31 is prime number)
The equation 2 times n equals 32, which is equivalent to the sum of the sum of n. This demonstrates that 16 is a super perfect number.
Now, let's examine another instance of a number that does not qualify as a superperfect integer.
Input:
N = 6
Output:
no
sig(6) = 1 + 2 + 3 + 6= 12
sig(12) = 1 + 2 + 3 + 4 +6 + 12 = 28
Multiplying 6 by 2 results in 12, which is not equal to 28. This demonstrates that 6 is not a super perfect number.
Code Implementation:
Let's consider a C program to verify if a given number is a superperfect number or not:-
#include<stdio.h>
int divisorsum(int n){
int sum = 0;
for (int a=1; a*a <= n; ++a){
if (n%a == 0) {
if (a == (n/a))
sum += a;
else
sum += (a + n/a);
}
}
return sum;
}
int main() {
int n = 64;
int n1 = divisorsum(n);
if(2*n == divisorsum(n1)){
printf("The number %d is a superperfect number", n);
} else{
printf("The number %d is not a superperfect number", n);
}
return 0;
}
Output:
[Program Output]
Explanation:
The sum of divisors of 64, denoted as σ(64), is calculated as 1 + 2 + 4 + 8 + 16 + 32 + 64, resulting in a value of 127. The divisors of 64 include 1, 2, 4, 8, 16, 32, and 64.
sig(127)=1+127=128(127 is a prime number)
2*n = 128 = sig(sig(n))
- This demonstrates that 64 qualifies as a superperfect number.
Complexity Analysis:
Time Complexity:
It has the time complexity of O(sqrt(n)).
Space Complexity:
It has the space complexity of O(1).