Rename Project Euler directories and other dependent changes (#3300)

* Rename all Project Euler directories:

Reason:
The change was done to maintain consistency throughout the directory
and to keep all directories in sorted order.

Due to the above change, some config files had to be modified:
'problem_22` -> `problem_022`

* Update scripts to pad zeroes in PE directories

project_euler/problem_014/sol2.py

@@ -0,0 +1,62 @@
+"""
+Problem 14: https://projecteuler.net/problem=14
+
+Collatz conjecture: start with any positive integer n. Next term obtained from
+the previous term as follows:
+
+If the previous term is even, the next term is one half the previous term.
+If the previous term is odd, the next term is 3 times the previous term plus 1.
+The conjecture states the sequence will always reach 1 regardless of starting
+n.
+
+Problem Statement:
+The following iterative sequence is defined for the set of positive integers:
+
+    n → n/2 (n is even)
+    n → 3n + 1 (n is odd)
+
+Using the rule above and starting with 13, we generate the following sequence:
+
+    13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
+
+It can be seen that this sequence (starting at 13 and finishing at 1) contains
+10 terms. Although it has not been proved yet (Collatz Problem), it is thought
+that all starting numbers finish at 1.
+
+Which starting number, under one million, produces the longest chain?
+"""
+from typing import List
+
+
+def collatz_sequence(n: int) -> List[int]:
+    """Returns the Collatz sequence for n."""
+    sequence = [n]
+    while n != 1:
+        if n % 2 == 0:
+            n //= 2
+        else:
+            n = 3 * n + 1
+        sequence.append(n)
+    return sequence
+
+
+def solution(n: int = 1000000) -> int:
+    """Returns the number under n that generates the longest Collatz sequence.
+
+    # The code below has been commented due to slow execution affecting Travis.
+    # >>> solution(1000000)
+    # 837799
+    >>> solution(200)
+    171
+    >>> solution(5000)
+    3711
+    >>> solution(15000)
+    13255
+    """
+
+    result = max([(len(collatz_sequence(i)), i) for i in range(1, n)])
+    return result[1]
+
+
+if __name__ == "__main__":
+    print(solution(int(input().strip())))