A336272 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.

17, 26, 120, 370, 392, 567, 680, 697, 847, 1066, 1089, 1183, 1233, 1299, 1371, 1448, 1904, 2009, 2169, 2176, 2281, 2307, 2535, 2600, 2619, 2785, 2845, 2993, 3150, 3370, 3825, 3944, 3983, 4035, 4095, 4290, 4706, 4760, 4879, 4905, 5655, 5811, 5835, 6137, 6375, 6570, 6936, 7202, 7913, 7995

Offset

1,1

Comments

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

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

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

Links

Hugo Pfoertner, Table of n, a(n) for n = 1..79

Sascha Kurz, On the generation of Heronian triangles, arXiv:1401.6150 [math.NT], 11 Jan 2014.

Sascha Kurz, On the generation of Heronian triangles, Serdica Journal of Computing Vol. 2 (2008), Pages 181-196.

Hugo Pfoertner, List of triangle sides, 20000 > i > j > k.

Example

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.
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].

Maple

# find all square Heron triangles whose longest side is between small and big
small:=1: big:=700:
A336272:=[]:triangles:=[]:
areasq16:=(a+b+c)*(a+b-c)*(a-b+c)*(-a+b+c):
# a>=b>=c
for a from small to big do:
  for b from ceil((a+1)/2) to a do:
    for c from a-b+1 to b do:
      if issqr(areasq16) and issqr(sqrt(areasq16)) and igcd(a, b, c)=1 then
        A336272:=[op(A336272), a]:
        triangles:=[op(triangles), [a, b, c]]:
      end if:
    od:
  od:
od: A336272; triangles;

Prog

(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

Crossrefs

Cf. A046128, A046129, A046130, A046131, A055592, A055593, A055594, A055595.

Sequence in context: A031204 A085051 A171954 * A154277 A140150 A166658

Adjacent sequences: A336269 A336270 A336271 * A336273 A336274 A336275

Keyword

nonn

Author

James R. Buddenhagen, Jul 15 2020

Extensions

a(42)-a(50) from Hugo Pfoertner, Jul 18 2020

Status

approved