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A367737 Ordered product of the terms in a primitive Pythagorean quadruple (with repetitions).

%I A367737 #7 Feb 16 2025 08:34:06
%S A367737 12,252,288,1008,1188,1872,2052,2100,2448,2772,3300,8400,8448,9108,
%T A367737 9828,11628,12768,13500,14688,17100,17388,17388,17472,18900,25500,
%U A367737 27900,29568,29568,31968,32292,32508,33408,50388,51612,54000,58212,58812,60372,62100,62832,63072,65472,65892,69300,69972
%N A367737 Ordered product of the terms in a primitive Pythagorean quadruple (with repetitions).
%C A367737 Every primitive Pythagorean quadruple (PPQ) generates a distinct Heronian triangle. This sequence is the area of such a triangle. If a, b, c, d form a PPQ where a^2 + b^2 + c^2 = d^2 it generates a primitive Heronian triangle whose three sides are b^2 + c^2, a^2 + c^2, a^2 + b^2. Its semiperimeter is d^2 and its area is a*b*c*d. It has an inradius and three exradii as a*b*c/d, b*c*d/a, a*c*d/b, a*b*d/c respectively.
%C A367737 a(n) == 0 mod 12.
%C A367737 A210484 is a subsequence because an integer Soddyian triangle has area m^2n^2(m+n)^2(m^2+mn+n^2) and semiperimeter (m^2+mn+n^2)^2 = m^2*n^2 + n^2(m+n)^2 + m^2(m+n)^2 where m >= n and GCD(m,n) = 1.  This is a PPQ.
%H A367737 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanQuadruple.html">Pythagorean Quadruple</a>.
%H A367737 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pythagorean_quadruple">Pythagorean quadruple</a>.
%e A367737 a(5)=1188 because the 5th occurrence of a PPQ sorted by the product of its term is (2, 6, 9, 11) and 1188 = 11*9*6*2.
%t A367737 lst = {}; Do[lst=Join[lst,Select[PowersRepresentations[k^2, 3, 2],Times@@#!=0&&GCD@@#==1 &]], {k, 1, 100}]; lst1=Sort@(Table[{a, b, c}=lst[[n]]; a*b*c*Sqrt[a^2+b^2+c^2], {n, 1, Length@lst}]); lst1[[1;;50]]
%Y A367737 Cf. A063011, A210484.
%K A367737 nonn
%O A367737 1,1
%A A367737 _Frank M Jackson_, Nov 28 2023