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
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
KEYWORD
nonn
AUTHOR
James R. Buddenhagen, Jul 15 2020
EXTENSIONS
a(42)-a(50) from Hugo Pfoertner, Jul 18 2020
STATUS
approved