Perfect Number

A number is a perfect number if it is equal to the sum of its proper divisors, that is, the sum of its positive divisors excluding the number itself. Find whether a given positive integer n is perfect or not.
Examples:

Input: n = 15
Output: false
Explanation: Divisors of 15 are 1, 3 and 5. Sum of divisors is 9 which is not equal to 15.

Input: n = 6
Output: true
Explanation: Divisors of 6 are 1, 2 and 3. Sum of divisors is 6.

[Naive Approach] Divisor Sum Method - O(n) Time and O(1) Space

A simple Solution is to go through every number from 1 to n-1 and check if it is a divisor. Maintain sum of all divisors. If sum becomes equal to n, then return true, else return false.

false

[Expected Approach] Optimized Divisor Search Method - O(sqrt n) Time and O(1) Space

An efficient Solution is to go through numbers till square root of n. If a number 'i' divides n, then add both 'i' and 'n/i' .

false

Interesting facts About Perfect Number
1) Every even perfect number is of the form 2^p−1⋅(2^p−1) , where 2^p−1 is a prime number.
2) It is unknown whether there are any odd perfect numbers.