**π‘ Problem Formulation:** In Python, working with integers that power up your computations can be both necessary and challenging. One common task is finding integers that can be expressed as x raised to the power of y, where x and y are also integers. For example, given an integer like 8, the desired output would be a confirmation that it is a powerful integer since it can be expressed as 2^3.

## Method 1: Using a Nested Loop

This method involves iterating through possible combinations of x and y within a certain range to check if they produce the target number when x is raised to the power of y (`x**y`

). It’s a brute force approach that’s understandable and straightforward for small ranges.

Here’s an example:

def is_powerful_integer(target): for x in range(1, int(target**0.5) + 1): for y in range(2, int(target**0.5) + 1): if x**y == target: return True return False print(is_powerful_integer(8))

Output: True

This code snippet defines a function `is_powerful_integer()`

that takes a target integer and checks if it is a powerful integer. It iterates over a range up to the square root of the target because any larger values will be unnecessary to check for the given power. If a combination of x and y is found to match the target when x is raised to y, it returns True.

## Method 2: Utilizing Python’s Math Library

The Python `math`

library provides functions like `sqrt()`

and `floor()`

which can be used to optimize the search by establishing tighter bounds. This method is more efficient than the brute force approach for larger numbers.

Here’s an example:

import math def is_powerful_integer(target): for x in range(1, math.floor(math.sqrt(target)) + 1): y = 2 while x**y <= target: if x**y == target: return True y += 1 return False print(is_powerful_integer(64))

Output: True

In this code sample, the `is_powerful_integer()`

function makes use of the `math.sqrt()`

and `math.floor()`

methods to refine the search range. It also replaces the inner loop with a while loop, increasing `y`

until `x**y`

exceeds the target. This is generally more efficient for finding powerful integers.

## Method 3: Hashing with Sets for Duplicate Avoidance

Preventing duplicate checks can be achieved by storing already computed powers in a set. A set in Python does not allow duplicates and has constant time complexity on average for insertions and lookups, making it suitable for this purpose.

Here’s an example:

def is_powerful_integer(target, max_base): seen = set() for x in range(2, max_base+1): power = x * x while power <= target: seen.add(power) power *= x return target in seen print(is_powerful_integer(16, 4))

Output: True

The function `is_powerful_integer()`

here takes a target and a max base to consider. It uses a set `seen`

to store all the powers it has computed. By checking if the target is in the set, it avoids unnecessary repeated checking and improves overall efficiency.

## Method 4: Using List Comprehensions and Ranges

List comprehensions in Python are a concise way to create lists. They can be combined with the `range()`

function to generate a list of powerful integers directly and check if a target is within this list.

Here’s an example:

def generate_powerful_integers(max_base): return [x**y for x in range(2, max_base+1) for y in range(2, int(math.log(max_base, x)) + 1)] powerful_integers = generate_powerful_integers(10) print(powerful_integers)

Output: [4, 8, 9, 16, 25, 27, 36, 49, 64, 81, 100]

This snippet creates a function `generate_powerful_integers()`

that generates a list of powerful integers up to a provided max base. The list comprehension iterates over the possible bases and their powers within the calculated range, including the logarithm to determine the maximum exponent for each base.

## Bonus One-Liner Method 5: Using itertools.product()

The `itertools.product()`

method from Python’s itertools library can be used to create a cartesian product of possible bases and exponents, directly mapping to the powerful integers without the need for nesting loops.

Here’s an example:

import itertools import math max_base = 10 powerful_integers = {x**y for x, y in itertools.product(range(2, max_base+1), repeat=2)} print(powerful_integers)

Output: {4, 8, 9, 16, 25, 27, 36, 49, 64, 81, 100}

This elegant one-liner uses a set comprehension along with `itertools.product()`

to construct the cartesian product of the range with itself, effectively simplifying the creation of the powerful integers set significantly.

## Summary/Discussion

**Method 1:**Nested Loop. It’s easy to understand and implement. However, it’s not efficient for large numbers or ranges due to its brute force nature.**Method 2:**Math Library. Utilizing the`math`

library can optimize the range of search and improve performance. It’s faster than a brute force approach but still involves iteration.**Method 3:**Hashing with Sets. By avoiding duplicate computations, this method is more efficient for a larger number of computations. However, it requires additional memory to store the set.**Method 4:**List Comprehensions. Offers a readable one-liner to generate the list of powerful integers. It is less memory efficient since it generates a list of all possible values.**Bonus Method 5:**itertools.product(). Provides a concise and readable way to generate powerful integers. This method is ideal for creating a set of unique powerful integers, but may not be as intuitive to those unfamiliar with itertools.