Add Proth number to maths (#5246)

* Add Proth number to maths

* Add test for 0 and more informative output

* Fixing test failure issue - unused variable

* Update proth_number.py

Co-authored-by: Christian Clauss <cclauss@me.com>

maths/proth_number.py

@@ -0,0 +1,77 @@
+"""
+Calculate the nth Proth number
+
+Source:
+    https://handwiki.org/wiki/Proth_number
+"""
+
+import math
+
+
+def proth(number: int) -> int:
+    """
+    :param number: nth number to calculate in the sequence
+    :return: the nth number in Proth number
+
+    Note: indexing starts at 1 i.e. proth(1) gives the first Proth number of 3
+
+    >>> proth(6)
+    25
+
+    >>> proth(0)
+    Traceback (most recent call last):
+    ...
+    ValueError: Input value of [number=0] must be > 0
+
+    >>> proth(-1)
+    Traceback (most recent call last):
+    ...
+    ValueError: Input value of [number=-1] must be > 0
+
+    >>> proth(6.0)
+    Traceback (most recent call last):
+    ...
+    TypeError: Input value of [number=6.0] must be an integer
+    """
+
+    if not isinstance(number, int):
+        raise TypeError(f"Input value of [number={number}] must be an integer")
+
+    if number < 1:
+        raise ValueError(f"Input value of [number={number}] must be > 0")
+    elif number == 1:
+        return 3
+    elif number == 2:
+        return 5
+    else:
+        block_index = number // 3
+        """
+        +1 for binary starting at 0 i.e. 2^0, 2^1, etc.
+        +1 to start the sequence at the 3rd Proth number
+        Hence, we have a +2 in the below statement
+        """
+        block_index = math.log(block_index, 2) + 2
+        block_index = int(block_index)
+
+        proth_list = [3, 5]
+        proth_index = 2
+        increment = 3
+        for block in range(1, block_index):
+            for move in range(increment):
+                proth_list.append(2 ** (block + 1) + proth_list[proth_index - 1])
+                proth_index += 1
+            increment *= 2
+
+    return proth_list[number - 1]
+
+
+if __name__ == "__main__":
+    for number in range(11):
+        value = 0
+        try:
+            value = proth(number)
+        except ValueError:
+            print(f"ValueError: there is no {number}th Proth number")
+            continue
+
+        print(f"The {number}th Proth number: {value}")