# Solution to Project Euler Problem 23: Non-abundant sums - A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number. A number n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n. As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24. By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers. However, this upper limit cannot be reduced any further by analysis even though it is known that the greatest number that cannot be expressed as the sum of two abundant numbers is less than this limit. Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

**Updated:**May 29, 2023 —

**Training Time:**3 minutes

Overseen by: Archangel Macsika

**Topic:**Project Euler Problem 23: Non-abundant sums.

**Difficulty:** Easy.

**Objective:** A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.

A number n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n.

As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24. By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers. However, this upper limit cannot be reduced any further by analysis even though it is known that the greatest number that cannot be expressed as the sum of two abundant numbers is less than this limit.

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

**Input:** None.

**Expected Output:** 4179871.

### Sikademy Solution in Java Programming Language

```
package sikademy;
/**
*
* @author Archangel Macsika
* Copyright (c) Sikademy. All rights reserved
*/
public class SikademyEulerSolution {
private static final int LIMIT = 28123;
private boolean[] isAbundant = new boolean[LIMIT + 1];
public String run() {
// Compute look-up table
for (int i = 1; i < isAbundant.length; i++)
isAbundant[i] = isAbundant(i);
int sum = 0;
for (int i = 1; i <= LIMIT; i++) {
if (!isSumOf2Abundants(i))
sum += i;
}
return Integer.toString(sum);
}
private boolean isSumOf2Abundants(int n) {
for (int i = 0; i <= n; i++) {
if (isAbundant[i] && isAbundant[n - i])
return true;
}
return false;
}
private static boolean isAbundant(int n) {
if (n < 1)
throw new IllegalArgumentException();
int sum = 1; // Sum of factors less than n
int end = Library.sqrt(n);
for (int i = 2; i <= end; i++) {
if (n % i == 0)
sum += i + n / i;
}
if (end * end == n)
sum -= end;
return sum > n;
}
public static void main(String[] args) {
SikademyEulerSolution solution = new SikademyEulerSolution();
System.out.println(solution.run());
}
}
```

### Sikademy Solution in Python Programming Language

```
#
# @author Archangel Macsika
# Copyright (c) Sikademy. All rights reserved.
#
def compute():
LIMIT = 28124
divisorsum = [0] * LIMIT
for i in range(1, LIMIT):
for j in range(i * 2, LIMIT, i):
divisorsum[j] += i
abundantnums = [i for (i, x) in enumerate(divisorsum) if x > i]
expressible = [False] * LIMIT
for i in abundantnums:
for j in abundantnums:
if i + j < LIMIT:
expressible[i + j] = True
else:
break
ans = sum(i for (i, x) in enumerate(expressible) if not x)
return str(ans)
if __name__ == "__main__":
print(compute())
```