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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.

%I A268396 #29 Oct 13 2018 12:33:34
%S A268396 44,117,240,240,252,275,88,234,480,85,132,720,160,231,792,132,351,720,
%T A268396 140,480,693,480,504,550,176,468,960,170,264,1440,220,585,1200,720,
%U A268396 756,825,320,462,1584,264,702,1440,280,960,1386,187,1020,1584,308,819,1680
%N 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.
%C A268396 Sides in increasing order of perimeter (a+b+c), where a < b < c.
%C A268396 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 A268396 Consider the set S(n) = {a(3*n-2), a(3*n-1), a(3*n)}. Then:
%C A268396 - at least one number in the set is divisible by 5
%C A268396 - at least one number in the set is divisible by 9
%C A268396 - at least one number in the set is divisible by 11
%C A268396 - at least one number in the set is divisible by 16
%C A268396 - at least two numbers in the set are divisible by 3
%C A268396 - at least two numbers in the set are divisible by 4.
%C A268396 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 A268396 Eli Maor, The Pythagorean Theorem: A 4,000-Year History, 2007, Princeton University Press, p. 134.
%H A268396 Wikipedia, <a href="http://en.wikipedia.org/wiki/Cuboid">Cuboid</a>
%H A268396 Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler_brick">Euler brick</a>
%Y A268396 Cf. A195816.
%Y A268396 See A245616 for a very similar sequence.
%Y A268396 Cf. A031173, A031174, A031175.
%K A268396 nonn
%O A268396 1,1
%A A268396 _Arkadiusz Wesolowski_, Feb 03 2016