Step by Step solution about How to resolve the algorithm Numbers which are the cube roots of the product of their proper divisors step by step in the Ruby programming language

This Ruby code is designed to identify and print a sequence of numbers known as the A111398 sequence. Here's a detailed explanation:

1. Library Import:

   require 'prime'

   This line imports the 'prime' library, which provides methods for working with prime numbers.

2. tau Function:

   def tau(n) = n.prime_division.inject(1){|res, (d, exp)| res *= exp+1}

   The 'tau' function calculates the number of divisors of a given number 'n'. It uses the 'prime_division' method to decompose 'n' into its prime factors and then applies the formula for the number of divisors.

3. A111398 Sequence Generation:

   a111398 = [1].chain (1..).lazy.select{|n| tau(n) == 8}

   Here, the A111398 sequence is generated. It starts with the number 1 and continues indefinitely, with each subsequent number being the next integer that has exactly 8 divisors. The 'chain' method is used to lazily generate the sequence, and the 'select' method applies the 'tau' function as a filter.

4. Printing the First 50 Numbers of the A111398 Sequence:

   puts "The first 50 numbers which are the cube roots of the products of their proper divisors:"
   p a111398.first(50)

   This section prints the first 50 numbers of the A111398 sequence. The 'first' method is used to select the first 50 elements, and the 'p' method prints them in a structured format.

5. Printing Specific Elements of the A111398 Sequence:

   [500, 5000, 50000].each{|n| puts "#{n}th: #{a111398.drop(n-1).next}" }

   This section prints the 500th, 5000th, and 50000th elements of the A111398 sequence. The 'drop' method is used to skip the first 'n-1' elements, and the 'next' method fetches the next element in the sequence.

require 'prime'

def tau(n) = n.prime_division.inject(1){|res, (d, exp)| res *= exp+1}
a111398 = [1].chain (1..).lazy.select{|n| tau(n) == 8}

puts "The first 50 numbers which are the cube roots of the products of their proper divisors:"
p a111398.first(50)
[500, 5000, 50000].each{|n| puts "#{n}th: #{a111398.drop(n-1).next}" }