# [Golang] Non-abundant sums - Problem 23 - Project Euler

Problem: 

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.

Solution:

4179871

Run Code on Go Playground

```package main

import (
"fmt"
)

// Get all prime factors of a given number n
func PrimeFactors(n int) (pfs []int) {
// Get the number of 2s that divide n
for n%2 == 0 {
pfs = append(pfs, 2)
n = n / 2
}

// n must be odd at this point. so we can skip one element
// (note i = i + 2)
for i := 3; i*i <= n; i = i + 2 {
// while i divides n, append i and divide n
for n%i == 0 {
pfs = append(pfs, i)
n = n / i
}
}

// This condition is to handle the case when n is a prime number
// greater than 2
if n > 2 {
pfs = append(pfs, n)
}

return
}

// return p^i
func Power(p, i int) int {
result := 1
for j := 0; j < i; j++ {
result *= p
}
return result
}

// formula comes from https://math.stackexchange.com/a/22723
func SumOfProperDivisors(n int) int {
pfs := PrimeFactors(n)

// key: prime
// value: prime exponents
m := make(map[int]int)
for _, prime := range pfs {
_, ok := m[prime]
if ok {
m[prime] += 1
} else {
m[prime] = 1
}
}

sumOfAllFactors := 1
for prime, exponents := range m {
sumOfAllFactors *= (Power(prime, exponents+1) - 1) / (prime - 1)
}
return sumOfAllFactors - n
}

func main() {
// Get all abundant numbers under 28123
var abundantNumbers []int
for i := 1; i <= 28123; i++ {
if SumOfProperDivisors(i) > i {
abundantNumbers = append(abundantNumbers, i)
}
}
//fmt.Println(abundantNumbers)

sumOfTwoAbundantNumbers := make(map[int]bool)
for i := 0; i < len(abundantNumbers); i++ {
for j := i; j < len(abundantNumbers); j++ {
a := abundantNumbers[i]
b := abundantNumbers[j]
if a+b <= 28123 {
//fmt.Println(a+b, " = ", a, " + ", b)
sumOfTwoAbundantNumbers[a+b] = true
} else {
break
}
}
}

sum := 0
for i := 1; i <= 28123; i++ {
_, ok := sumOfTwoAbundantNumbers[i]
if !ok {
//fmt.Println(i)
sum += i
}
}
fmt.Println(sum)
}
```

Tested on: Go Playground

References: