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%I #20 Dec 11 2017 05:35:54
%S 1381,2931,5156,58658,70135,79012,89680,106966,152084,171416,197522,
%T 212885,266098,295306,400078,434790,675720,789403,863969,866606,
%U 917338,936413,1085618,1149892,1242687,1432297,1628115,2116668,2241911,2250397,2418925,2694694,2699343,3022126,3036895,3059130
%N The ordered integer image of the 1-to-1 mapping of primitive Heronian triples (PHT) into the integers using Cantor's pairing function for triples (N^3 -> N).
%C This mapping of the Heronian triple (a,b,c) to an integer is unique and uses Cantor's pairing function K(i,j) = (i+j)(i+j+1)/2+j so that (a,b,c) -> K(K(a,b),c). The table of PHT's used to generate the sequence was obtained from lists generated by _Sascha Kurz_ (see Link). The list contains a triple for every possible PHT with a maximum side length of 10000. The triples are in the form (a,b,c) where a >= b >= c and where a <= 10000.
%H Sascha Kurz, <a href="http://hdl.handle.net/10525/382">On the generation of Heronian triangles</a>, Serdica Journal of Computing. 2 (2) (2008): pp. 181-196
%H Sascha Kurz, <a href="http://www.wm-archive.uni-bayreuth.de/index.php?id=554&L=3">Lists of primitive Heronian triples</a>, Bayreuth University
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pairing_function#Cantor_pairing_function">Cantor's pairing function</a>, and <a href="http://en.wikipedia.org/wiki/Heronian_triangle">Heronian triangle</a>
%e A PHT with sides (a,b,c) = (21,20,13) maps to K(K(21,20),13) = K(881,13) = 400078 = a(15), where Cantor's pairing function K is simply A001477 in its two-argument tabular form A001477(k, n) = n + (k+n)*(k+n+1)/2.
%e A PHT with sides (a,b,c) = (29,21,20) maps to K(K(29,21),20) = 866606 = a(20). This is a primitive Pythagorean triangle (thus also a primitive Heronian triangle), listed as term a(5)=33 in A277557.
%t Cantor[i_, j_] := (i+j)(i+j+1)/2+j; nn=50; lst1=ReadList["C:/primitive_heronian_triangles_1_10000.txt", {Number, Number, Number}]; lst2=Select[lst1, #[[1]]<=2 nn &]; lst={}; Do[({a, b, c}=lst2[[n]]; k=Cantor[Cantor[a, b], c]; AppendTo[lst, k]), {n, 1, Length[lst2]}]; Sort[Select[lst, #<Cantor[Cantor[nn, nn], nn] &]] (* For download of file of primitive Heronian triples see Link *)
%Y Cf. A001477, A243808, A277557.
%K nonn
%O 1,1
%A _Frank M Jackson_, Oct 24 2016
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