|
| 1 | +""" |
| 2 | +Find all narcissistic numbers up to a given limit using dynamic programming. |
| 3 | +
|
| 4 | +A narcissistic number (also known as an Armstrong number or plus perfect number) |
| 5 | +is a number that is the sum of its own digits each raised to the power of the |
| 6 | +number of digits. |
| 7 | +
|
| 8 | +For example, 153 is a narcissistic number because 153 = 1^3 + 5^3 + 3^3. |
| 9 | +
|
| 10 | +This implementation uses dynamic programming with memoization to efficiently |
| 11 | +compute digit powers and find all narcissistic numbers up to a specified limit. |
| 12 | +
|
| 13 | +The DP optimization caches digit^power calculations. When searching through many |
| 14 | +numbers, the same digit power calculations occur repeatedly (e.g., 153, 351, 135 |
| 15 | +all need 1^3, 5^3, 3^3). Memoization avoids these redundant calculations. |
| 16 | +
|
| 17 | +Examples of narcissistic numbers: |
| 18 | + Single digit: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 |
| 19 | + Three digit: 153, 370, 371, 407 |
| 20 | + Four digit: 1634, 8208, 9474 |
| 21 | + Five digit: 54748, 92727, 93084 |
| 22 | +
|
| 23 | +Reference: https://en.wikipedia.org/wiki/Narcissistic_number |
| 24 | +""" |
| 25 | + |
| 26 | + |
| 27 | +def find_narcissistic_numbers(limit: int) -> list[int]: |
| 28 | + """ |
| 29 | + Find all narcissistic numbers up to the given limit using dynamic programming. |
| 30 | +
|
| 31 | + This function uses memoization to cache digit power calculations, avoiding |
| 32 | + redundant computations across different numbers with the same digit count. |
| 33 | +
|
| 34 | + Args: |
| 35 | + limit: The upper bound for searching narcissistic numbers (exclusive) |
| 36 | +
|
| 37 | + Returns: |
| 38 | + list[int]: A sorted list of all narcissistic numbers below the limit |
| 39 | +
|
| 40 | + Examples: |
| 41 | + >>> find_narcissistic_numbers(10) |
| 42 | + [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] |
| 43 | + >>> find_narcissistic_numbers(160) |
| 44 | + [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153] |
| 45 | + >>> find_narcissistic_numbers(400) |
| 46 | + [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371] |
| 47 | + >>> find_narcissistic_numbers(1000) |
| 48 | + [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407] |
| 49 | + >>> find_narcissistic_numbers(10000) |
| 50 | + [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474] |
| 51 | + >>> find_narcissistic_numbers(1) |
| 52 | + [0] |
| 53 | + >>> find_narcissistic_numbers(0) |
| 54 | + [] |
| 55 | + """ |
| 56 | + if limit <= 0: |
| 57 | + return [] |
| 58 | + |
| 59 | + narcissistic_nums = [] |
| 60 | + |
| 61 | + # Memoization: cache[(power, digit)] = digit^power |
| 62 | + # This avoids recalculating the same power for different numbers |
| 63 | + power_cache: dict[tuple[int, int], int] = {} |
| 64 | + |
| 65 | + def get_digit_power(digit: int, power: int) -> int: |
| 66 | + """Get digit^power using memoization (DP optimization).""" |
| 67 | + if (power, digit) not in power_cache: |
| 68 | + power_cache[(power, digit)] = digit**power |
| 69 | + return power_cache[(power, digit)] |
| 70 | + |
| 71 | + # Check each number up to the limit |
| 72 | + for number in range(limit): |
| 73 | + # Count digits |
| 74 | + num_digits = len(str(number)) |
| 75 | + |
| 76 | + # Calculate sum of powered digits using memoized powers |
| 77 | + remaining = number |
| 78 | + digit_sum = 0 |
| 79 | + while remaining > 0: |
| 80 | + digit = remaining % 10 |
| 81 | + digit_sum += get_digit_power(digit, num_digits) |
| 82 | + remaining //= 10 |
| 83 | + |
| 84 | + # Check if narcissistic |
| 85 | + if digit_sum == number: |
| 86 | + narcissistic_nums.append(number) |
| 87 | + |
| 88 | + return narcissistic_nums |
| 89 | + |
| 90 | + |
| 91 | +if __name__ == "__main__": |
| 92 | + import doctest |
| 93 | + |
| 94 | + doctest.testmod() |
| 95 | + |
| 96 | + # Demonstrate the dynamic programming approach |
| 97 | + print("Finding all narcissistic numbers up to 10000:") |
| 98 | + print("(Using memoization to cache digit power calculations)") |
| 99 | + print() |
| 100 | + |
| 101 | + narcissistic_numbers = find_narcissistic_numbers(10000) |
| 102 | + print(f"Found {len(narcissistic_numbers)} narcissistic numbers:") |
| 103 | + print(narcissistic_numbers) |
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