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

The Python code you provided is an implementation of the A005179 sequence, which represents the smallest number with exactly n divisors. The code includes two different implementations of the sequence: one using a generator function (sequence) and the other using a more functional programming approach (a005179). The code also includes a helper function, divisors, which computes the divisors of a given number, and several utility functions for working with sequences and numbers (take, until, properDivisors, primeFactors, and go).

Here's a detailed explanation of how the code works:

divisors function:

• This function takes an integer n and returns a list of its divisors.
• It starts by initializing a list divs with the number 1 (which is always a divisor of any number).
• Then, it iterates through the numbers from 2 to the square root of n (rounded up to the nearest integer).
• For each number ii, it checks if it evenly divides n. If it does, it adds ii and n / ii to the divs list.
• Finally, it adds n to the divs list and returns the set of all unique divisors in the list.

sequence function:

• This is a generator function that yields the smallest number with exactly n divisors, for each n in the range [1, max_n].
• It initializes a variable n to 0 and enters a while loop that runs indefinitely.
• Inside the loop, it increments n by 1.
• It then initializes another variable ii to 0 and enters a nested while loop.
• Inside the nested loop, it increments ii by 1.
• It computes the divisors of ii using the divisors function and checks if the length of the divisors list is equal to n.
• If the condition is met, it yields ii and breaks out of the nested loop.
• The outer while loop continues to iterate, incrementing n and repeating the process for the next value of n.

a005179 function:

• This is a generator function that yields the A005179 sequence.
• It uses a comprehension to generate the sequence. The comprehension starts with a generator that produces the numbers 1, 2, 3, ... indefinitely.
• It then applies a filter to the generator, which checks if the number n (the next number in the sequence) satisfies the condition that it has exactly n divisors. The condition is checked using the properDivisors function, which returns the proper divisors of a number (excluding the number itself).
• The filtered generator is then used to create the sequence, which is yielded by the a005179 function.

properDivisors function:

• This function computes the proper divisors of an integer n, which are the divisors of n excluding n itself.
• It uses a recursive helper function go to compute the divisors. The go function takes two arguments: a list of divisors a and a list of prime factors x.
• The go function first initializes a list qr with the divisors a and the product of the elements in x.
• It then maps the elements of qr to a list of products of the elements in a and the accumulated product of the elements in x.
• The resulting list is sorted and then returned, excluding the last element (which is n).

primeFactors function:

• This function computes the prime factors of an integer n.
• It uses a recursive helper function f to compute the prime factors. The f function takes a tuple qr as an argument, where qr[0] is a prime factor and qr[1] is the quotient of n divided by the prime factor.
• The f function returns a tuple qr where qr[0] is the next prime factor and qr[1] is the quotient of n divided by the next prime factor.
• The f function is applied recursively until the quotient qr[1] is less than or equal to the square root of n.
• The resulting list of prime factors is returned.

take function:

• This is a utility function that takes a number n and a sequence xs as arguments and returns the prefix of xs of length n, or xs itself if n is greater than the length of xs.

until function:

• This is a utility function that takes a predicate function p and a function f as arguments and returns a function that applies f to an initial value x until p(x) is True. The resulting value of x is returned.

The main function uses the a005179 function to generate the first 15 terms of the A005179 sequence and print them to the console.

def divisors(n):
    divs = [1]
    for ii in range(2, int(n ** 0.5) + 3):
        if n % ii == 0:
            divs.append(ii)
            divs.append(int(n / ii))
    divs.append(n)
    return list(set(divs))


def sequence(max_n=None):
    n = 0
    while True:
        n += 1
        ii = 0
        if max_n is not None:
            if n > max_n:
                break
        while True:
            ii += 1
            if len(divisors(ii)) == n:
                yield ii
                break


if __name__ == '__main__':
    for item in sequence(15):
        print(item)


'''Smallest number with exactly n divisors'''

from itertools import accumulate, chain, count, groupby, islice, product
from functools import reduce
from math import sqrt, floor
from operator import mul


# a005179 :: () -> [Int]
def a005179():
    '''Integer sequence: smallest number with exactly n divisors.'''
    return (
        next(
            x for x in count(1)
            if n == 1 + len(properDivisors(x))
        ) for n in count(1)
    )


# --------------------------TEST---------------------------
# main :: IO ()
def main():
    '''First 15 terms of a005179'''
    print(main.__doc__ + ':\n')
    print(
        take(15)(
            a005179()
        )
    )


# -------------------------GENERIC-------------------------

# properDivisors :: Int -> [Int]
def properDivisors(n):
    '''The ordered divisors of n, excluding n itself.
    '''
    def go(a, x):
        return [a * b for a, b in product(
            a,
            accumulate(chain([1], x), mul)
        )]
    return sorted(
        reduce(go, [
            list(g) for _, g in groupby(primeFactors(n))
        ], [1])
    )[:-1] if 1 < n else []


# primeFactors :: Int -> [Int]
def primeFactors(n):
    '''A list of the prime factors of n.
    '''
    def f(qr):
        r = qr[1]
        return step(r), 1 + r

    def step(x):
        return 1 + (x << 2) - ((x >> 1) << 1)

    def go(x):
        root = floor(sqrt(x))

        def p(qr):
            q = qr[0]
            return root < q or 0 == (x % q)

        q = until(p)(f)(
            (2 if 0 == x % 2 else 3, 1)
        )[0]
        return [x] if q > root else [q] + go(x // q)

    return go(n)


# take :: Int -> [a] -> [a]
# take :: Int -> String -> String
def take(n):
    '''The prefix of xs of length n,
       or xs itself if n > length xs.
    '''
    return lambda xs: (
        xs[0:n]
        if isinstance(xs, (list, tuple))
        else list(islice(xs, n))
    )


# until :: (a -> Bool) -> (a -> a) -> a -> a
def until(p):
    '''The result of repeatedly applying f until p holds.
       The initial seed value is x.
    '''
    def go(f, x):
        v = x
        while not p(v):
            v = f(v)
        return v
    return lambda f: lambda x: go(f, x)


# MAIN ---
if __name__ == '__main__':
    main()