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%I #36 Jan 28 2023 12:13:45
%S 2491,3616,4385,4450,4783,4982,7232,7473,7974,8770,8900,9566,9964,
%T 10848,11784,12455,12503,13155,13350,13565,14086,14141,14349,14464,
%U 14778,14946,15948,16389,17394,17437,17540,17800,18080,19132,19453,19928,21696,21925,22250,22419,22821,23568,23915,23922,24079
%N Sums of Pythagorean sextuples in increasing order: The sums of sets of six natural numbers which correspond to the lengths of the edges of a tetrahedron whose four faces are all different Pythagorean triangles.
%C A Pythagorean sextuple contains four Pythagorean triples.
%C The sequence is sorted on increasing sum of set.
%C See attached a-file to see the specific values in the sets.
%C The first sum shared by two sets is 956670.
%C The first set contains the values 104, 153, 672, 185, 680, 697.
%C The first set to have a ratio less than 2:1 between its highest and lowest value is 1680, 1925, 2052, 2555, 2652, 3277.
%C The method used for finding the sets was to connect two Pythagorean triangles along one edge and then calculate the length of the remaining edge.
%H Andreas Boe, <a href="/A248548/b248548.txt">Table of n, a(n) for n = 1..2426</a>
%H Andreas Boe, <a href="/A248548/a248548.txt">Sets of Pythagorean sextuples, cross indexed with the base sequence</a>
%F Sums a + b + c + d + e + f where a^2 + b^2 = d^2, a^2 + c^2 = e^2, b^2 + e^2 = f^2, c^2 + d^2 = f^2.
%e The first value in the sequence: 2491.
%e 104^2 + 153^2 = 185^2, 104^2 + 672^2 = 680^2, 153^2 + 680^2 = 697^2, 672^2 + 185^2 = 697^2;
%e 104 + 153 + 185 + 672 + 680 + 697 = 2491.
%K nonn
%O 1,1
%A _Andreas Boe_, Oct 08 2014
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