Here is an observation regarding the mathematical representation of the Kabbalistic concept of fourteen Tikkunim, i.e., regarding the exhaustive list of factorials n! such that the smallest square number that is not smaller than n!, exceeds n! by a square number. This list is as follows:
0! + 0^2 = 1^2
1! + 0^2 = 1^2
4! + 1^2 = 5^2
5! + 1^2 = 11^2
6! + 3^2 = 27^2
7! + 1^2 = 71^2
8! + 9^2 = 201^2
9! + 27^2 = 603^2
10! + 15^2 = 1905^2
11! + 18^2 = 6318^2
13! + 288^2 = 78912^2
14! + 420^2 = 295260^2
15! + 464^2 = 1143536^2
16! + 1856^2 = 4574144^2
The following computation also takes fourteen steps and also leads to the two final numbers 1143536 and 4574144. Each step, except the first, takes the result of the previous step and multiplies it with 3, 2, 4, 8, or 16, or adds 1, or subtracts 1.
0 = 0
0 + 1 = 1
1 * 3 = 3
3 * 8 = 24
24 - 1 = 23
23 * 4 = 92
92 + 1 = 93
93 * 16 = 1488
1488 + 1 = 1489
1489 * 3 = 4467
4467 * 16 = 71472
71472 - 1 = 71471
71471 * 16 = 1143536
1143536 * 4 = 4574144
