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## Python: Check if a Number is a Narcissistic Number In number theory, a **Narcissistic number** (also known as a **pluperfect digital invariant (PPDI)**, an **Armstrong number**, or a **plus perfect number**) is a number that is the sum of its own digits each raised to the power of the number of digits. For example, $153$ is a 3-digit narcissistic number because: $$1^3 + 5^3 + 3^3 = 1 + 125 + 27 = 153$$ Similarly, $1634$ is a 4-digit narcissistic number because: $$1^4 + 6^4 + 3^4 + 4^4 = 1 + 1296 + 81 + 256 = 1634$$ This tutorial demonstrates how to implement a clean, efficient Python program to check whether a given integer is a Narcissistic number. --- ## Implementation Approach To determine if a number is narcissistic, we can follow these steps: 1. Convert the integer to a string to easily access each individual digit and determine the total number of digits ($n$). 2. Iterate through each digit, convert it back to an integer, and raise it to the power of $n$. 3. Sum these powered values. 4. Compare the final sum with the original number. If they are equal, the number is narcissistic. --- ## Code Example Below is the standard Python implementation using a helper function: ```python def is_narcissistic_number(num): # Convert the number to a string to process digit by digit num_str = str(num) # Get the total number of digits (n) n = len(num_str) # Calculate the sum of each digit raised to the power of n sum_of_powers = sum(int(digit) ** n for digit in num_str) # Return True if the sum equals the original number, otherwise False return sum_of_powers == num # --- Test the function --- number = 153 if is_narcissistic_number(number): print(f"{number} is a Narcissistic number.") else: print(f"{number} is not a Narcissistic number.") ``` ### Output ```text 153 is a Narcissistic number. ``` --- ## Code Explanation 1. **`is_narcissistic_number(num)`**: This function accepts an integer `num` as its parameter. 2. **`num_str = str(num)`**: Converts the integer to a string representation. This allows us to easily iterate over each digit and use Python's built-in sequence operations. 3. **`n = len(num_str)`**: Determines the length of the string, which represents the number of digits ($n$) in the original integer. 4. **`sum_of_powers = sum(int(digit) ** n for digit in num_str)`**: Uses a generator expression to iterate through each character in the string, cast it back to an integer, raise it to the power of `n`, and sum the results using the built-in `sum()` function. 5. **`return sum_of_powers == num`**: Performs a boolean comparison. If the calculated sum matches the input number, it returns `True`; otherwise, it returns `False`. --- ## Advanced Use Case: Finding Narcissistic Numbers in a Range If you want to find all Narcissistic numbers within a specific range (for example, all 3-digit Narcissistic numbers between 100 and 999), you can use the helper function inside a loop or list comprehension: ```python # Find all 3-digit Narcissistic numbers narcissistic_3_digits = [x for x in range(100, 1000) if is_narcissistic_number(x)] print("3-digit Narcissistic numbers:", narcissistic_3_digits) ``` ### Output ```text 3-digit Narcissistic numbers: [153, 370, 371, 407] ``` --- ## Performance Considerations * **String Conversion vs. Mathematical Extraction**: Converting the number to a string is highly readable and idiomatic in Python. However, for extremely performance-critical applications or low-level environments, extracting digits mathematically using modulo (`%`) and floor division (`//`) operators can avoid the overhead of string allocation. * **Time Complexity**: $\mathcal{O}(d)$ where $d$ is the number of digits in the integer. Since $d = \lfloor\log_{10}(num)\rfloor + 1$, the algorithm runs in logarithmic time relative to the value of the input number, making it highly efficient.