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

12, 252, 288, 1008, 1188, 1872, 2052, 2100, 2448, 2772, 3300, 8400, 8448, 9108, 9828, 11628, 12768, 13500, 14688, 17100, 17388, 17388, 17472, 18900, 25500, 27900, 29568, 29568, 31968, 32292, 32508, 33408, 50388, 51612, 54000, 58212, 58812, 60372, 62100, 62832, 63072, 65472, 65892, 69300, 69972

Offset: 1

Author

Frank M Jackson, Nov 28 2023

Keywords

• nonn

Comments

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.

a(n) == 0 mod 12.

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.

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Pythagorean Quadruple.
• Wikipedia, Pythagorean quadruple.

Crossrefs

Cf. A063011, A210484.

Programs

Mathematica

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]]