A268396 Sides of Pythagorean cuboids: triples (a, b, c) that are integral length sides of a rectangular cuboid for which the three face diagonals x, y, z also have integral length.

44, 117, 240, 240, 252, 275, 88, 234, 480, 85, 132, 720, 160, 231, 792, 132, 351, 720, 140, 480, 693, 480, 504, 550, 176, 468, 960, 170, 264, 1440, 220, 585, 1200, 720, 756, 825, 320, 462, 1584, 264, 702, 1440, 280, 960, 1386, 187, 1020, 1584, 308, 819, 1680

Offset: 1

Author

Arkadiusz Wesolowski, Feb 03 2016

Keywords

• nonn

Comments

Sides in increasing order of perimeter (a+b+c), where a < b < c.

A triple (a, b, c) of integers belongs to this sequence if and only if all of the numbers sqrt(a^2 + b^2), sqrt(b^2 + c^2), and sqrt(a^2 + c^2) are also integers.

Consider the set S(n) = {a(3*n-2), a(3*n-1), a(3*n)}. Then:

- at least one number in the set is divisible by 5

- at least one number in the set is divisible by 9

- at least one number in the set is divisible by 11

- at least one number in the set is divisible by 16

- at least two numbers in the set are divisible by 3

- at least two numbers in the set are divisible by 4.

The list of "Sides of ..." is A195816, while this sequence lists "Triples...", i.e., (a(3n-2), a(3n-1), a(3n)) = (A031175(k), A031174(k), A031173(k)) for some k, n >= 1. (The order is not the same as for A031173 etc, e.g., the 5th through 8th triple have decreasing largest sides.) Also, in A031173, A031174, A031175 and others, the side naming convention is a > b > c, the opposite of here. - M. F. Hasler, Oct 11 2018

References

• Eli Maor, The Pythagorean Theorem: A 4,000-Year History, 2007, Princeton University Press, p. 134.

Links

• Wikipedia, Cuboid
• Wikipedia, Euler brick

Crossrefs

Cf. A195816.

See A245616 for a very similar sequence.

Cf. A031173, A031174, A031175.