Step by Step solution about How to resolve the algorithm Amicable pairs step by step in the Go programming language

The provided Go program finds and prints amicable pairs below 20,000. Here's a detailed explanation:

1. amicable pairs: Two numbers are called amicable pairs if the sum of the proper divisors (positive divisors excluding the number itself) of one number is equal to the other number, and vice versa. For example, (220, 284) is an amicable pair because the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, and they sum up to 284. Similarly, the proper divisors of 284 are 1, 2, 4, 71, and 142, and they sum up to 220.

2. pfacSum function: This function takes an integer i as input and returns the sum of its proper divisors. It iterates from 1 to i/2 and checks if p divides i. If it does, it adds p to the sum. This effectively sums up the proper divisors of i.

3. main function:

   • Array Initialization: The main function initializes an array a of size 20,000. This array will store the sum of proper divisors for numbers from 1 to 19,999.
   • Populating the Array: It uses a loop to populate the array a with the sum of proper divisors for each number from 1 to 19,999. The pfacSum function is used to calculate the sum of proper divisors.
   • Printing Amicable Pairs: After populating the array, the program enters another loop to search for amicable pairs. It iterates from 2 to 19,999 (excluding 1). For each number n, it retrieves the sum of proper divisors from the array a. If this sum m is greater than n (to avoid printing the same pair twice), less than 20,000 (to stay within the array bounds), and equal to n's sum of proper divisors stored in a[m], then (n, m) is an amicable pair. The program then prints these amicable pairs.

In summary, this program efficiently finds and prints amicable pairs below 20,000. It uses an array to store the sum of proper divisors for each number from 1 to 19,999 to quickly determine amicable pairs.

package main

import "fmt"

func pfacSum(i int) int {
    sum := 0
    for p := 1; p <= i/2; p++ {
        if i%p == 0 {
            sum += p
        }
    }
    return sum
}

func main() {
    var a[20000]int
    for i := 1; i < 20000; i++ {
        a[i] = pfacSum(i)
    }
    fmt.Println("The amicable pairs below 20,000 are:")
    for n := 2; n < 19999; n++ {
        m := a[n]
        if m > n && m < 20000 && n == a[m] {
            fmt.Printf("  %5d and %5d\n", n, m)
        } 
    }
}