#### Proth Number

```"""
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:
"""
+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 = int(math.log(number // 3, 2)) + 2

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__":
import doctest

doctest.testmod()

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}")
```  