A277632 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).

1381, 2931, 5156, 58658, 70135, 79012, 89680, 106966, 152084, 171416, 197522, 212885, 266098, 295306, 400078, 434790, 675720, 789403, 863969, 866606, 917338, 936413, 1085618, 1149892, 1242687, 1432297, 1628115, 2116668, 2241911, 2250397, 2418925, 2694694, 2699343, 3022126, 3036895, 3059130

Offset: 1

Author

Frank M Jackson, Oct 24 2016

Keywords

• nonn

Comments

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.

Examples

			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.
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.
		

Links

• Sascha Kurz, On the generation of Heronian triangles, Serdica Journal of Computing. 2 (2) (2008): pp. 181-196
• Sascha Kurz, Lists of primitive Heronian triples, Bayreuth University
• Wikipedia, Cantor's pairing function, and Heronian triangle

Crossrefs

Cf. A001477, A243808, A277557.

Programs

Mathematica

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, #