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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.
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%I #29 Oct 13 2018 12:33:34

%S 44,117,240,240,252,275,88,234,480,85,132,720,160,231,792,132,351,720,

%T 140,480,693,480,504,550,176,468,960,170,264,1440,220,585,1200,720,

%U 756,825,320,462,1584,264,702,1440,280,960,1386,187,1020,1584,308,819,1680

%N 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.

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

%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.

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

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

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

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

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

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

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

%C 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

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

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Cuboid">Cuboid</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler_brick">Euler brick</a>

%Y Cf. A195816.

%Y See A245616 for a very similar sequence.

%Y Cf. A031173, A031174, A031175.

%K nonn

%O 1,1

%A _Arkadiusz Wesolowski_, Feb 03 2016