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 A224302 Sorted perimeters of primitive Heronian triangles. 1
 12, 16, 18, 30, 32, 36, 36, 40, 42, 42, 44, 48, 50, 50, 54, 54, 54, 56, 60, 64, 64, 64, 66, 68, 70, 70, 72, 76, 78, 80, 80, 84, 84, 84, 84, 84, 90, 90, 90, 96, 98, 98, 98, 98, 98, 98, 100, 100, 100, 104, 104, 108, 108, 108, 108, 108, 110, 112, 112, 112, 112 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Here a primitive Heronian triangle has integer sides a,b,c with gcd(a,b,c) = 1 and integral area. The perimeters of primitive Heronian triangles are even [Wenzel Šimerka, 1869]. - Mo Li, Feb 02 2020 REFERENCES L. E. Dickson, History of the Theory of Numbers, vol. II: Diophantine Analysis, Dover, 2005, p. 196. [21a] LINKS Giovanni Resta, Table of n, a(n) for n = 1..10002 L. E. Dickson, History of the Theory of Numbers, vol. II, 1952, see p. 196 [21 a]. Michael Somos, Heronian Triangle Table Wikipedia, Heronian triangle EXAMPLE a(1) = 12 as it is the perimeter of the Heronian triangle having sides 3,4,5 and is the smallest Heronian triangle perimeter. a(2) = 16 as it is the perimeter of the Heronian triangle having sides 5,5,6 and is the 2nd smallest Heronian triangle perimeter. MATHEMATICA hQ[a_, b_, c_] := IntegerQ@Sqrt@Block[{s = (a + b + c)/2}, s (s - a) (s - b) (s - c)]; Sort[Reap[Do[If[GCD[a, b, c] == 1 && hQ[a, b, c], Sow@(a + b + c)], {a, 100}, {b, a}, {c, a - b + 1, b}]][[2, 1]]] (* The last numbers given may not be exactly in the right place. *) (* Jinyuan Wang, Feb 02 2020 *) CROSSREFS Cf. A120131, A120132, A120133, A224301. Sequence in context: A356863 A112548 A032620 * A096468 A334983 A334984 Adjacent sequences: A224299 A224300 A224301 * A224303 A224304 A224305 KEYWORD nonn AUTHOR Mihir Mathur, Apr 04 2013 EXTENSIONS Corrected and extended by Giovanni Resta, Apr 04 2013 STATUS approved

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Last modified September 18 20:35 EDT 2024. Contains 376002 sequences. (Running on oeis4.)