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A096467 Numbers that can be the longest side of a primitive Heronian triangle. 3
5, 6, 8, 13, 15, 17, 20, 21, 24, 25, 26, 28, 29, 30, 35, 36, 37, 39, 40, 41, 42, 44, 45, 48, 50, 51, 52, 53, 55, 56, 58, 60, 61, 63, 65, 66, 68, 69, 70, 73, 74, 75, 77, 80, 82, 85, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 112, 113 (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. Note that all primes of the form 4k+1 are in this sequence. It appears that a prime of the form 4k+3 is never the longest side of a Heronian triangle. Cheney's article contains many theorems about these triangles.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..240

Wm. Fitch Cheney, Jr., Heronian Triangles, Amer. Math. Monthly, Vol. 36, No. 1 (Jan 1929), 22-28.

Eric Weisstein's World of Mathematics, Heronian Triangle

EXAMPLE

5 is on this list because the triangle with sides 3, 4, 5 has integral area.

MATHEMATICA

nn=150; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s] && GCD[a, b, c]==1, area2=s(s-a)(s-b)(s-c); If[area2>0 && IntegerQ[Sqrt[area2]], AppendTo[lst, a]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

CROSSREFS

Cf. A083875 (area/6 of primitive Heronian triangles), A096468 (perimeter of primitive Heronian triangles).

Cf. A239246.

Sequence in context: A188054 A276374 A184803 * A323041 A105830 A067527

Adjacent sequences:  A096464 A096465 A096466 * A096468 A096469 A096470

KEYWORD

nonn

AUTHOR

T. D. Noe, Jun 22 2004

STATUS

approved

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Last modified February 19 08:57 EST 2019. Contains 320309 sequences. (Running on oeis4.)