Add narcissistic number finder with dynamic programming (#13971)

* Add narcissistic number finder with dynamic programming

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* Update dynamic_programming/narcissistic_number.py

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* Update dynamic_programming/narcissistic_number.py

Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>

* Update dynamic_programming/narcissistic_number.py

Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>

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Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>

dynamic_programming/narcissistic_number.py

@@ -0,0 +1,103 @@
+"""
+Find all narcissistic numbers up to a given limit using dynamic programming.
+
+A narcissistic number (also known as an Armstrong number or 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 narcissistic number because 153 = 1^3 + 5^3 + 3^3.
+
+This implementation uses dynamic programming with memoization to efficiently
+compute digit powers and find all narcissistic numbers up to a specified limit.
+
+The DP optimization caches digit^power calculations. When searching through many
+numbers, the same digit power calculations occur repeatedly (e.g., 153, 351, 135
+all need 1^3, 5^3, 3^3). Memoization avoids these redundant calculations.
+
+Examples of narcissistic numbers:
+    Single digit: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
+    Three digit: 153, 370, 371, 407
+    Four digit: 1634, 8208, 9474
+    Five digit: 54748, 92727, 93084
+
+Reference: https://en.wikipedia.org/wiki/Narcissistic_number
+"""
+
+
+def find_narcissistic_numbers(limit: int) -> list[int]:
+    """
+    Find all narcissistic numbers up to the given limit using dynamic programming.
+
+    This function uses memoization to cache digit power calculations, avoiding
+    redundant computations across different numbers with the same digit count.
+
+    Args:
+        limit: The upper bound for searching narcissistic numbers (exclusive)
+
+    Returns:
+        list[int]: A sorted list of all narcissistic numbers below the limit
+
+    Examples:
+        >>> find_narcissistic_numbers(10)
+        [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
+        >>> find_narcissistic_numbers(160)
+        [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153]
+        >>> find_narcissistic_numbers(400)
+        [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371]
+        >>> find_narcissistic_numbers(1000)
+        [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407]
+        >>> find_narcissistic_numbers(10000)
+        [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474]
+        >>> find_narcissistic_numbers(1)
+        [0]
+        >>> find_narcissistic_numbers(0)
+        []
+    """
+    if limit <= 0:
+        return []
+
+    narcissistic_nums = []
+
+    # Memoization: cache[(power, digit)] = digit^power
+    # This avoids recalculating the same power for different numbers
+    power_cache: dict[tuple[int, int], int] = {}
+
+    def get_digit_power(digit: int, power: int) -> int:
+        """Get digit^power using memoization (DP optimization)."""
+        if (power, digit) not in power_cache:
+            power_cache[(power, digit)] = digit**power
+        return power_cache[(power, digit)]
+
+    # Check each number up to the limit
+    for number in range(limit):
+        # Count digits
+        num_digits = len(str(number))
+
+        # Calculate sum of powered digits using memoized powers
+        remaining = number
+        digit_sum = 0
+        while remaining > 0:
+            digit = remaining % 10
+            digit_sum += get_digit_power(digit, num_digits)
+            remaining //= 10
+
+        # Check if narcissistic
+        if digit_sum == number:
+            narcissistic_nums.append(number)
+
+    return narcissistic_nums
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
+    # Demonstrate the dynamic programming approach
+    print("Finding all narcissistic numbers up to 10000:")
+    print("(Using memoization to cache digit power calculations)")
+    print()
+
+    narcissistic_numbers = find_narcissistic_numbers(10000)
+    print(f"Found {len(narcissistic_numbers)} narcissistic numbers:")
+    print(narcissistic_numbers)