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

The provided Kotlin code defines functions to calculate the divisors of a number, determine if a number is abundant, determine if a number is semiperfect, perform a modified Sieve of Eratosthenes, and find the first 25 weird numbers. Here's a breakdown:

1. divisors Function:

   • Takes an integer n as input.
   • Initializes two mutable lists: divs to store found divisors and divs2 to store the complement of divisors.
   • Iterates from 2 to the square root of n to find divisors.
   • If n is divisible by i, both i and n/i are added to the divisors lists.
   • After the loop, divs2 is appended with the reverse of divs.
   • Returns the combined list of divisors.

2. abundant Function:

   • Takes an integer n and a list of its divisors divs.
   • Checks if the sum of the divisors is greater than n.
   • If so, returns true, indicating that the number is abundant. Otherwise, returns false.

3. semiperfect Function:

   • Takes an integer n and a list of its divisors divs.
   • Recursively determines if n can be expressed as the sum of distinct divisors.
   • The function explores different combinations and returns true if n is semiperfect. Otherwise, it returns false.

4. sieve Function:

   • Takes an integer limit as input.
   • Initializes a Boolean array w of size limit to track the status of numbers.
   • Iterates through even numbers from 2 to limit.
   • For each number i, it calculates its divisors and checks if it's abundant.
   • If i is abundant, it marks all its multiples as abundant and semiperfect in the w array.
   • Returns the Boolean array w.

5. main Function:

   • Calls the sieve function with a limit of 17000 and stores the result in the w array.
   • Initializes a counter count to 0 and a maximum count max to 25.
   • Prints a message about finding the first 25 weird numbers.
   • Iterates through even numbers n starting from 2.
   • For each n, it checks if it's not marked as abundant or semiperfect in the w array.
   • If n is weird, it prints it and increments the counter.
   • Once 25 weird numbers are found, it prints a newline.

The key concept of the code is to determine if a number is weird. In mathematics, a weird number is an even positive integer that is not abundant and not semiperfect. The code uses a slightly modified Sieve of Eratosthenes to mark all abundant and semiperfect numbers up to a limit, and then it iterates through even numbers to find the first 25 weird numbers.

// Version 1.3.21

fun divisors(n: Int): List<Int> {
    val divs = mutableListOf(1)
    val divs2 = mutableListOf<Int>()
    var i = 2
    while (i * i <= n) {
        if (n % i == 0) {
            val j = n / i
            divs.add(i)
            if (i != j) divs2.add(j)
        }
        i++
    }
    divs2.addAll(divs.asReversed())
    return divs2
}

fun abundant(n: Int, divs: List<Int>) = divs.sum() > n

fun semiperfect(n: Int, divs: List<Int>): Boolean {
    if (divs.size > 0) {
        val h = divs[0]
        val t = divs.subList(1, divs.size)
        if (n < h) {
            return semiperfect(n, t)
        } else {
            return n == h || semiperfect(n-h, t) || semiperfect(n, t)
        }
    } else {
        return false
    }
}

fun sieve(limit: Int): BooleanArray {
    // false denotes abundant and not semi-perfect.
    // Only interested in even numbers >= 2
    val w = BooleanArray(limit)
    for (i in 2 until limit step 2) {
        if (w[i]) continue
        val divs = divisors(i)
        if (!abundant(i, divs)) {
            w[i] = true
        } else if (semiperfect(i, divs)) {
            for (j in i until limit step i) w[j] = true
        }
    }
    return w
}

fun main() {
    val w = sieve(17000)
    var count = 0
    val max = 25
    println("The first 25 weird numbers are:")
    var n = 2
    while (count < max) {
        if (!w[n]) {
            print("$n ")
            count++
        }
        n += 2
    }
    println()
}