The Boolean Pythagorean triples problem, a $200$-terabyte proof, and $d=163$

Question

I came across this interesting math article,

"Computer cracks 200-terabyte maths proof"

where one phrase caught my attention and I quote, "... all triples could be multi-coloured in integers up to $7824$". Alternatively, from page 2 of this paper,

Theorem 1. The set {$1,\dots,7824$} can be partitioned into two parts, such that no part contains a Pythagorean triple. This is impossible for {$1,\dots,7825$}.

The number $N=7824$ was awfully familiar. A quick factorization showed that it was in fact,

$$N = 7824 = 2^4 \times 3\times \color{blue}{163}$$

Questions:

1. Does anybody know why the largest Heegner number $163$ figures in the largest $N$ that can be multi-colored in the Boolean Pythagorean triples problem?
2. A272709 is the sequence $2, 4, 8, 16, 24, 48, 96, 192,....0,0,0,0,0\dots$ where the zeros start at $a(7825)$. What is the exact value of $a(7824)$? (In the comments, it just says $a(7824)\geq8$.)