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

Understanding the Code Structure

The code is divided into four main functions:

• chowla: Calculates the Chowla sum of a given integer.
• sieve: Uses the Sieve of Chowla to generate a list of composite numbers.
• commatize: Formats a large integer with thousand separators (commas).
• main: Runs the program.

Chowla Numbers (Function: chowla)

The chowla function calculates the Chowla sum of an integer n. The Chowla sum is the sum of all distinct divisors of n. For example, the Chowla sum of 12 is 28 (1 + 2 + 3 + 4 + 6 + 12).

The function loops through all integers from 2 to the square root of n, checking if n is divisible by each integer. If it is, it adds the corresponding divisors to the sum.

Sieve of Chowla (Function: sieve)

The sieve function uses the Sieve of Chowla to generate a list of composite numbers. It creates a boolean array c, where true indicates a composite number and false indicates a prime number. The sieve starts with odd numbers greater than or equal to 3.

For each odd number i that is not composite (i.e., c[i] == false), the function checks if chowla(i) == 0. If this condition is true, it means i is a "Chowla number."

Chowla numbers have an interesting property: If i is a Chowla number, then all multiples of i (except for i itself) are composite. Therefore, the function marks all multiples of i as composite in the c array.

Commatizing Numbers (Function: commatize)

The commatize function converts a large integer into a string with thousand separators (commas).

Main Program (Function: main)

The main function is the entry point of the program. It does the following:

1. Prints the Chowla sums for integers from 1 to 37.
2. Uses the sieve function to generate a list of composite numbers up to a limit of 10 million.
3. Displays the count of prime numbers up to various powers of 10.
4. Searches for perfect numbers up to a limit of 35 million. A perfect number is a number that is equal to the sum of its proper divisors (divisors excluding the number itself). The function uses the Chowla sum to identify perfect numbers.

package main

import "fmt"

func chowla(n int) int {
    if n < 1 {
        panic("argument must be a positive integer")
    }
    sum := 0
    for i := 2; i*i <= n; i++ {
        if n%i == 0 {
            j := n / i
            if i == j {
                sum += i
            } else {
                sum += i + j
            }
        }
    }
    return sum
}

func sieve(limit int) []bool {
    // True denotes composite, false denotes prime.
    // Only interested in odd numbers >= 3
    c := make([]bool, limit)
    for i := 3; i*3 < limit; i += 2 {
        if !c[i] && chowla(i) == 0 {
            for j := 3 * i; j < limit; j += 2 * i {
                c[j] = true
            }
        }
    }
    return c
}

func commatize(n int) string {
    s := fmt.Sprintf("%d", n)
    le := len(s)
    for i := le - 3; i >= 1; i -= 3 {
        s = s[0:i] + "," + s[i:]
    }
    return s
}

func main() {
    for i := 1; i <= 37; i++ {
        fmt.Printf("chowla(%2d) = %d\n", i, chowla(i))
    }
    fmt.Println()

    count := 1
    limit := int(1e7)
    c := sieve(limit)
    power := 100
    for i := 3; i < limit; i += 2 {
        if !c[i] {
            count++
        }
        if i == power-1 {
            fmt.Printf("Count of primes up to %-10s = %s\n", commatize(power), commatize(count))
            power *= 10
        }
    }

    fmt.Println()
    count = 0
    limit = 35000000
    for i := uint(2); ; i++ {
        p := 1 << (i - 1) * (1<<i - 1) // perfect numbers must be of this form
        if p > limit {
            break
        }
        if chowla(p) == p-1 {
            fmt.Printf("%s is a perfect number\n", commatize(p))
            count++
        }
    }
    fmt.Println("There are", count, "perfect numbers <= 35,000,000")
}