Pythagorean Numbers

Of course I knew about Pythagorean triples and even primitive Pythagorean triples but I hadn't heard of Pythagorean numbers. The term emerged when I was researching my daily number, 24570, using the OEIS. This number was paired with 24576 and the smaller followed the larger in sequence A228875: Pairs of Pythagorean numbers differing by 6. This difference is apparently the minimum possible. The sequence started:

24, 30, 54, 60, 210, 216, 330, 336, 480, 486, 540, 546, 720, 726, 750, 756, 1344, 1350, 1710, 1716, 2160, 2166, 8664, 8670, 8970, 8976, 10080, 10086, 10290, 10296, 12144, 12150, 15600, 15606, 18144, 18150, 24570, 24576, 28560, 28566, 30240, 30246, 34650, 34656

This didn't really explain what constituted a Pythagorean number. However, as I discovered here, the definition of such as number is that it is the area of a Pythagorean triangle and primitive Pythagorean number is the area of a primitive Pythagorean triangle. Sequence A009111 provides an ordered list the areas of Pythagorean triangles, effectively providing a list of the initial Pythagorean numbers. Oddly, 24570 turns out to be 294th and 295th in this list. The reason for this will soon become clear.

While I knew that 24570 was a Pythagorean number and thus the area of a Pythagorean triangle, I didn't know the integer sides that comprised such a triangle but it seemed that there were two possible triangles because the number occupied two positions in the list. It took a little fiddling around in WolframAlpha to come up with the numbers.

Thus the triangles were 84, 585, 591 and 180, 273, 327. The number 24570 is not a primitive Pythagorean number because the members of each triplet are divisible by three. The equivalent Pythagorean triplets are 28, 195, 197 and 60, 91, 109.