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%I #47 Jul 26 2020 18:56:30
%S 17,26,120,370,392,567,680,697,847,1066,1089,1183,1233,1299,1371,1448,
%T 1904,2009,2169,2176,2281,2307,2535,2600,2619,2785,2845,2993,3150,
%U 3370,3825,3944,3983,4035,4095,4290,4706,4760,4879,4905,5655,5811,5835,6137,6375,6570,6936,7202,7913,7995
%N Length of longest side of a primitive square Heron triangle, i.e., a triangle with relatively prime integer sides and area the square of a positive integer.
%C The triangle [a(23)=2535, 2329, 544] with gcd(2329, 544) = 17 is the first square Heron triangle for which the 3 sides [i, j, k] are not pairwise coprime, i.e., max(gcd(i,j), gcd(i,k), gcd(j,k)) > 1, but gcd(i,j,k) = 1. Are there more square Heron triangles with this property? - _Hugo Pfoertner_, Jul 18 2020
%C There are other square Heron triangles with this property, e.g. [a(31)=3825, 2704, 1921] with gcd(1921, 3825) = 17; [a(??)=41460721, 38639097, 17536520] with gcd(38639097, 17536520) = 41; [a(??)=153915025, 139641489, 25224736] with gcd(25224736, 153915025) = 17; and [a(??)=4325561361, 3459908000, 1430190961] with gcd(3459908000, 1430190961) = 73. - _James R. Buddenhagen_, Jul 20 2020
%C Terms are given with multiplicity, e.g. if there are two primitive square Heron triangles with equal longest sides, that longest side is listed as a term of the sequence twice (this is very rare). - _James R. Buddenhagen_, Jul 21 2020
%H Hugo Pfoertner, <a href="/A336272/b336272.txt">Table of n, a(n) for n = 1..79</a>
%H Sascha Kurz, <a href="https://arxiv.org/abs/1401.6150">On the generation of Heronian triangles</a>, arXiv:1401.6150 [math.NT], 11 Jan 2014.
%H Sascha Kurz, <a href="http://hdl.handle.net/10525/382">On the generation of Heronian triangles</a>, Serdica Journal of Computing Vol. 2 (2008), Pages 181-196.
%H Hugo Pfoertner, <a href="/A336272/a336272_1.txt">List of triangle sides</a>, 20000 > i > j > k.
%e 17 is in the sequence because the triangle with sides [17, 10, 9] has longest side 17 and area 6^2, the square of a positive integer; 26 is in the sequence because the triangle with sides [26, 25, 3] has longest side 26 and has area 6^2, the square of a positive integer.
%e Triangles with sides [a, b, c] corresponding to the first 8 terms of this sequence are: [17, 10, 9], [26, 25, 3], [120, 113, 17], [370, 357, 41], [392, 353, 255], [567, 424, 305], [680, 441, 337], [697, 657, 104].
%p # find all square Heron triangles whose longest side is between small and big
%p small:=1: big:=700:
%p A336272:=[]:triangles:=[]:
%p areasq16:=(a+b+c)*(a+b-c)*(a-b+c)*(-a+b+c):
%p # a>=b>=c
%p for a from small to big do:
%p for b from ceil((a+1)/2) to a do:
%p for c from a-b+1 to b do:
%p if issqr(areasq16) and issqr(sqrt(areasq16)) and igcd(a,b,c)=1 then
%p A336272:=[op(A336272),a]:
%p triangles:=[op(triangles),[a,b,c]]:
%p end if:
%p od:
%p od:
%p od: A336272;triangles;
%o (PARI) for(a=1,1200,for(b=ceil((a+1)/2),a,for(c=a-b+1,b,if(gcd([a,b,c])==1,if(ispower((a+b+c)*(a+b-c)*(a-b+c)*(b+c-a),4),print1(a,", ")))))) \\ _Hugo Pfoertner_, Jul 18 2020
%Y Cf. A046128, A046129, A046130, A046131, A055592, A055593, A055594, A055595.
%K nonn
%O 1,1
%A _James R. Buddenhagen_, Jul 15 2020
%E a(42)-a(50) from _Hugo Pfoertner_, Jul 18 2020