Step by Step solution about How to resolve the algorithm Weird numbers step by step in the Go programming language

Overview:

This program identifies and prints the first 25 "weird" numbers, which are semiperfect even numbers that are neither abundant (sum of divisors is greater than the number) nor semiperfect (sum of a subset of divisors equals the number).

Key Concepts:

Divisors:

• The divisors function computes the divisors of a positive integer n.

Abundant Numbers:

• The abundant function checks if a number n is abundant, meaning the sum of its divisors is greater than n.

Semiperfect Numbers:

• The semiperfect function checks if a number n is semiperfect, meaning it can be written as the sum of some subset of its divisors.

Sieve:

• The sieve function implements the Sieve of Eratosthenes algorithm to identify weird numbers up to a given limit.

Main Function:

• The main function calls sieve to generate a sieve that marks abundant and not semiperfect numbers as false.
• It iterates through even numbers starting from 2 and prints the first 25 numbers that are not marked as abundant and not semiperfect (i.e., weird numbers).

Walkthrough:

1. The divisors function uses a for loop to find the divisors of n. It starts at 2 and checks up to the square root of n.
2. The abundant function sums the divisors of n and returns true if the sum is greater than n.
3. The semiperfect function uses a recursive algorithm to determine if n can be represented as the sum of some subset of its divisors. It considers the first divisor and recursively explores different combinations.
4. The sieve function initializes a slice of booleans w with a length equal to the given limit. It then iterates through even numbers and computes their divisors.
   • If a number is not abundant, it's marked as true in the sieve.
   • If a number is semiperfect, it marks multiples of that number as true.
5. In the main function, the program generates the sieve and then looks for the first 25 weird numbers (not marked as abundant or not semiperfect) and prints them.

Example:

For the first few weird numbers (up to 25):

Input: limit = 17000

Output:

The first 25 weird numbers are:
74 836 1210 1964 2204 2264 2534 2924 3124 3644 3904 4076 4376 4636 5104 5426 5636 5894 6464 6904 7364 7684 8254 8324 8534

package main

import "fmt"

func divisors(n int) []int {
    divs := []int{1}
    divs2 := []int{}
    for i := 2; i*i <= n; i++ {
        if n%i == 0 {
            j := n / i
            divs = append(divs, i)
            if i != j {
                divs2 = append(divs2, j)
            }
        }
    }
    for i := len(divs) - 1; i >= 0; i-- {
        divs2 = append(divs2, divs[i])
    }
    return divs2
}

func abundant(n int, divs []int) bool {
    sum := 0
    for _, div := range divs {
        sum += div
    }
    return sum > n
}

func semiperfect(n int, divs []int) bool {
    le := len(divs)
    if le > 0 {
        h := divs[0]
        t := divs[1:]
        if n < h {
            return semiperfect(n, t)
        } else {
            return n == h || semiperfect(n-h, t) || semiperfect(n, t)
        }
    } else {
        return false
    }
} 

func sieve(limit int) []bool {
    // false denotes abundant and not semi-perfect.
    // Only interested in even numbers >= 2
    w := make([]bool, limit)
    for i := 2; i < limit; i += 2 {
        if w[i] {
            continue
        }
        divs := divisors(i)
        if !abundant(i, divs) {
            w[i] = true
        } else if semiperfect(i, divs) {
            for j := i; j < limit; j += i {
                w[j] = true
            }
        }
    }
    return w
}

func main() {
    w := sieve(17000)
    count := 0
    const max = 25
    fmt.Println("The first 25 weird numbers are:")
    for n := 2; count < max; n += 2 {
        if !w[n] {
            fmt.Printf("%d ", n)
            count++
        }
    }
    fmt.Println()
}