Step by Step solution about How to resolve the algorithm Sequence: nth number with exactly n divisors step by step in the Haskell programming language

The provided Haskell code snippet calculates and prints a list of pairs of integers (n, d), where n is an integer between 1 and 35, and d is the n-th divisor of n.

The code first defines a function calc that takes an integer as an argument and returns a list of pairs of integers. Each pair in the list consists of an integer x and the number of divisors of x. The function calc uses the divisorCount function from the Math.NumberTheory.ArithmeticFunctions module to count the number of divisors of an integer.

The code then defines a function havingNthDivisors that takes an integer as an argument and returns a list of pairs of integers. Each pair in the list consists of an integer n and the number of divisors of n. The function havingNthDivisors uses the calc function to calculate the number of divisors of each integer n and then filters the list to only include the pairs where the number of divisors is equal to n.

The code then defines a function nths that returns a list of pairs of integers. Each pair in the list consists of an integer n and the n-th divisor of n. The function nths uses the isPrime function from the Math.NumberTheory.Primes.Testing module to check if an integer is prime. If an integer is prime, its n-th divisor is n itself. Otherwise, the n-th divisor of n is the first element of the list returned by the havingNthDivisors function applied to n.

The code then defines a function f that takes an integer as an argument and returns the n-th divisor of n. The function f is used in the nths function to calculate the n-th divisor of an integer.

The code then defines a function nthPrime that takes an integer as an argument and returns the n-th prime number. The function nthPrime uses the toEnum and unPrime functions from the Math.NumberTheory.Primes module to convert between an integer and a Prime value.

The code then defines the main function, which is the entry point of the program. The main function calls the mapM_ function from the Control.Monad module to print each pair in the list returned by the nths function.

import           Control.Monad                         (guard)
import           Math.NumberTheory.ArithmeticFunctions (divisorCount)
import           Math.NumberTheory.Primes              (Prime, unPrime)
import           Math.NumberTheory.Primes.Testing      (isPrime)

calc :: Integer -> [(Integer, Integer)]
calc n = do
  x <- [1..]
  guard (even n || odd n && f x == x)
  [(x, divisorCount x)]
 where f n = floor (sqrt $ realToFrac n) ^ 2

havingNthDivisors :: Integer -> [(Integer, Integer)]
havingNthDivisors n = filter ((==n) . snd) $ calc n

nths :: [(Integer, Integer)]
nths = do
  n <- [1..35] :: [Integer]
  if isPrime n then
    pure (n, nthPrime (fromIntegral n) ^ pred n)
  else
    pure (n, f n)
 where
  f n = fst (havingNthDivisors n !! pred (fromIntegral n))
  nthPrime n = unPrime (toEnum n :: Prime Integer)

main :: IO ()
main = mapM_ print nths