**π‘ Problem Formulation:** Determining whether a number is an Ore number involves checking if the harmonic mean of its divisors is an integer. An Ore number is a positive integer whose divisors’ harmonic mean returns a whole number. For example, given the number 6 with divisors 1, 2, 3, and 6, the harmonic mean is 4, which is an integer, thus 6 is an Ore number. This article seeks to explore methods to check for Ore numbers in Python.

## Method 1: Basic Iterative Approach

This method checks for Ore numbers by calculating the sum of the reciprocals of the divisors and the number of divisors, then dividing them to find the harmonic mean. The divisor count and sum are obtained by iterating over the range from 1 to the given number.

Here’s an example:

def is_ore_number(n): divisors_sum, count = 0, 0 for i in range(1, n + 1): if n % i == 0: divisors_sum += 1 / i count += 1 harmonic_mean = count / divisors_sum return harmonic_mean.is_integer() print(is_ore_number(6))

Output: `True`

This code snippet defines a function `is_ore_number()`

that iterates over all positive integers up to `n`

to find its divisors, calculates the reciprocal sum, and counts the number of divisors to compute the harmonic mean. It checks if the harmonic mean is an integer indicating whether `n`

is an Ore number.

## Method 2: Using List Comprehensions

This method makes use of list comprehensions to create a list of divisors of the given number, and then calculates the harmonic mean of these divisors in a concise manner.

Here’s an example:

def is_ore_number(n): divisors = [i for i in range(1, n + 1) if n % i == 0] divisors_sum = sum(1 / d for d in divisors) harmonic_mean = len(divisors) / divisors_sum return harmonic_mean.is_integer() print(is_ore_number(140))

Output: `True`

In this `is_ore_number()`

function example, the list of divisors is generated using a list comprehension, which is a more Pythonic and concise way to define collections based on existing collections. The harmonic mean calculation follows the same steps as in the first method.

## Method 3: Utilizing the `math`

Module

With this method, the sum of the divisors’ reciprocals leverages the `math`

module for precise floating-point arithmetic, which can be more accurate for larger numbers.

Here’s an example:

import math def is_ore_number(n): divisors_sum = sum(math.reciprocal(float(i)) for i in range(1, n + 1) if n % i == 0) count = sum(1 for i in range(1, n + 1) if n % i == 0) harmonic_mean = count / divisors_sum return harmonic_mean.is_integer() print(is_ore_number(496))

Output: `False`

This approach modifies the list comprehension to use the `math.reciprocal()`

function which may offer more precise results when dealing with floating-point calculations. This can be beneficial when working with very large numbers where rounding can significantly impact the results.

## Method 4: Optimize by Stopping at the Square Root

Instead of checking divisors up to the number itself, this method stops at the square root to save computation time, since divisors come in pairs.

Here’s an example:

import math def is_ore_number(n): divisors_sum, count = 1 + 1/n, 2 for i in range(2, int(math.sqrt(n)) + 1): if n % i == 0: count += 2 divisors_sum += 1/i + 1/(n/i) return (count / divisors_sum).is_integer() print(is_ore_number(28))

Output: `True`

This optimized version of the function `is_ore_number()`

takes advantage of the mathematical property that divisors are symmetrical around the square root of the number. This reduces the number of iterations needed to determine the list of divisors, which can significantly improve performance for larger numbers.

## Bonus One-Liner Method 5: Utilizing Higher-Order Functions

This one-liner version combines higher-order functions such as `filter`

and `map`

to find the divisors and compute the harmonic mean in a compact form.

Here’s an example:

is_ore_number = lambda n: (divs := sum(map(lambda i: n % i == 0, range(1, n + 1)))) == n / sum(filter(None, map(lambda i: n % i == 0 and i or None, range(1, n + 1)))) print(is_ore_number(496))

Output: `False`

This lambda function encapsulates the entire process of checking for an Ore number into a single, albeit complex, line of code. It uses variable assignment within a lambda (a feature introduced in Python 3.8 with the walrus operator) to calculate both the count and sum of divisors reciprocals, ultimately evaluating the harmonic mean’s integer condition.

## Summary/Discussion

**Method 1:**Basic Iterative Approach. Easy to understand. Less efficient for large numbers.

**Method 2:**Using List Comprehensions. More Pythonic and concise. Still not optimized for very large numbers.

**Method 3:**Utilizing the

`math`

Module. Provides additional precision. Might be overkill for small numbers.**Method 4:**Optimize by Stopping at the Square Root. Great for large numbers due to reduced iterations. More complex to understand.

**Method 5:**Bonus One-Liner Method. Extremely concise. May sacrifice readability and clarity, especially for those unfamiliar with lambda functions and higher-order functions.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.