What's Special About 204?

There's a scene in the first episode of "Prime Target" in which the student asks his supervising professor to find the pattern hidden inside the number (204) that he's written on the blackboard. Initially the professor is relucant but eventually he goes to the blackboard and writes that: $$204^2=23^3+24^3+25^3$$ I don't like this series for various reasons and didn't even get through all of the first episode. Before the student wrote on the blackboard, he sitting at the professor's desk and asks the question: "204 is a fascinating number. Don't you find it fascinating?". When the professor tries to get his student back on track (he is reviewing the work that the student has submitted), that's when the student approaches the blackboard and writes the number 204.

Now the professor's response to the student's goading is odd. While the arithmetic of what he's written is correct, the original question asks what pattern is hidden inside the number 204, not its square. Ignoring the number squared and considering just 204, there are a number of possible responses. Let's list them.

• 204 is a sum of all the perfect squares from 1 to 64. In other words:
  $1^2+2^2+ \dots +7^2+8^2 = 1+4+ \dots +49 +64=204$
• 204 is a square pyramidal number: 204 balls may be stacked in a pyramid whose base is an 8 × 8 square. See Figure 1.

Figure 1

• Both 204 and its square are sums of a pair of twin primes:
  $204 = 101 + 103 \text{ and } 204^2 = 41616 = 20807 + 20809$
  The only smaller numbers with the same property are 12 and 84. This property does involve the square of the number but the number itself shares the same property.*

There are other properties that are listed in the Wikipedia article but these three stand out. The far more obscure property of the square of the number being equal to the sum of three consecutive integers is shared by 3 and 36 since: $$3^2=9=0^3+1^3+2^3\\36=6^2=1^3+2^3+3^3$$ Some of the other properties of 204 are as follows:

• There are exactly 204 ways to place three non-attacking chess queens on a 5 × 5 board.
• There are exactly 204 squares of an infinite chess move that are eight knight's moves from the center.

Anyway, the point is that the choice of the property that the student is thinking of and the professor writes on the board is an odd choice.

* OEIS A213784  Numbers $k$ such that both $k$ and $k^2$ are sums of a twin prime pair. The initial members of this sequence are 12, 84, 204, 456, 1140, 5424, 10044, 11004, 13656, 17940, 27804, 36576.