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

1,1

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.

Links

Table of n, a(n) for n=1..45.

Eric Weisstein's World of Mathematics, Pythagorean Quadruple.

Wikipedia, Pythagorean quadruple.

Example

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.

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]]

Crossrefs

Cf. A063011, A210484.

Sequence in context: A245913 A245917 A265453 * A210484 A099139 A289565

Adjacent sequences: A367734 A367735 A367736 * A367738 A367739 A367740

Keyword

nonn

Author

Frank M Jackson, Nov 28 2023

Status

approved