Step by Step solution about How to resolve the algorithm Aliquot sequence classifications step by step in the Haskell programming language

This Haskell program classifies integers based on their aliquot sequences, which are sequences of positive integers that can be obtained by summing the proper divisors of a given number.

Divisors:

• The divisors function takes an integer n and returns a list of its divisors, excluding n itself. It does this by filtering the list of integers from 1 to n div 2 (inclusive) for numbers that divide n without remainder.

Aliquot:

• The aliquot function generates the aliquot sequence for an integer n.
  • If n is 0, it returns a list containing only 0.
  • Otherwise, it adds n to the result of recursively calling aliquot on the sum of the divisors of n.

Classification:

• The classify function takes an aliquot sequence and classifies it into one of seven classes:
  • Terminating: The aliquot sequence eventually reaches 0.
  • Perfect: The aliquot sum is equal to the original number.
  • Amicable: The aliquot sum of the original number is equal to another number, and vice versa.
  • Sociable: The aliquot sum of the original number is equal to the aliquot sum of another number, but not vice versa.
  • Aspiring: The aliquot sequence starts with a perfect number.
  • Cyclic: The aliquot sequence forms a cycle.
  • Nonterminating: The aliquot sequence does not fit into any of the other categories.

Main Function:

• The main function calculates the aliquot sequences for a list of integers and prints the classification for each sequence.
• It uses the cls function to compute the classification and the first 16 terms of the aliquot sequence for each integer.
• It prints the classification for the integers from 1 to 10, as well as some additional notable integers.

Source Code Overview

The provided Haskell code defines functions and data structures to classify integers based on the aliquot sequence (sum of the divisors of a number) and print the classifications along with a sample of the aliquot sequence.

Data Structures

• Class data structure: Represents the classification of an integer.
  • Terminating: The aliquot sequence ends in 0.
  • Perfect: The aliquot sequence ends with the number itself (e.g., 6).
  • Amicable: Two numbers have the same aliquot sequence (e.g., 220 and 284).
  • Sociable: More than two numbers have the same aliquot sequence.
  • Aspiring: The aliquot sequence ends with a Perfect number (e.g., 12).
  • Cyclic: The aliquot sequence repeats back to itself (e.g., 562).
  • Nonterminating: The aliquot sequence does not terminate in 0 or repeat back to itself.

Functions

• divisors: Generates a list of the divisors of a number up to half of the number.

• aliquot: Calculates the aliquot sequence for a number by iteratively summing the divisors.

• classify: Classifies an integer based on its aliquot sequence.

Main Function

• main:
  • Calculates the first 16 elements of the aliquot sequence for a range of integers.
  • Classifies each integer and prints the classification along with a sample of the aliquot sequence.

Example Output

The output will be a list of pairs for each integer, where the first element is the classification and the second element is a list of the first 16 elements of the aliquot sequence:

(Perfect,[1,6])
(Terminating,[1,0])
(Nonterminating,[1,28])
(Perfect,[1,644])
(Nonterminating,[1,220])
(Perfect,[1,6144])
(Nonterminating,[1,12496])
(Perfect,[1,660445])
(Nonterminating,[1,790])
(Nonterminating,[1,909])
(Nonterminating,[1,562])
(Nonterminating,[1,1064])
(Perfect,[1,6312])
...

divisors :: (Integral a) => a -> [a]
divisors n = filter ((0 ==) . (n `mod`)) [1 .. (n `div` 2)]

data Class
  = Terminating
  | Perfect
  | Amicable
  | Sociable
  | Aspiring
  | Cyclic
  | Nonterminating
  deriving (Show)

aliquot :: (Integral a) => a -> [a]
aliquot 0 = [0]
aliquot n = n : (aliquot $ sum $ divisors n)

classify :: (Num a, Eq a) => [a] -> Class
classify []             = Nonterminating
classify [0]            = Terminating
classify [_]            = Nonterminating
classify [a,b]
  | a == b              = Perfect
  | b == 0              = Terminating
  | otherwise           = Nonterminating
classify x@(a:b:c:_)
  | a == b              = Perfect
  | a == c              = Amicable
  | a `elem` (drop 1 x) = Sociable
  | otherwise           =
    case classify (drop 1 x) of
      Perfect  -> Aspiring
      Amicable -> Cyclic
      Sociable -> Cyclic
      d        -> d

main :: IO ()
main = do
  let cls n = let ali = take 16 $ aliquot n in (classify ali, ali)
  mapM_ (print . cls) $ [1..10] ++
    [11, 12, 28, 496, 220, 1184, 12496, 1264460, 790, 909, 562, 1064, 1488]