A072294 -id:A072294 - cp's OEIS frontend

A083875 One-sixth the areas of primitive Heronian triangles sorted.

1, 2, 4, 5, 6, 7, 10, 11, 12, 14, 15, 19, 20, 21, 22, 26, 28, 30, 33, 34, 35, 36, 39, 40, 42, 44, 45, 48, 50, 51, 55, 56, 60, 65, 66, 68, 70, 76, 77, 78, 80, 84, 85, 87, 88, 91, 95, 104, 105, 110, 114, 115, 119, 120, 124, 126, 130, 132, 133, 140, 144, 146

Offset: 1

Lekraj Beedassy, Jun 18 2003

nonn

The Mathematica program examines all primitive Heronian triangles having longest side <= 900, which guarantees that we have found all triangles with area/6 <= 150. - T. D. Noe, Jun 22 2004

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A072294.

nn=900; 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, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[Select[lst, #

More terms from T. D. Noe, Jun 22 2004

A046131 Areas of scalene integer Heronian triangles (A046128, A046129, A046130) sorted by increasing c and b.

Original entry on oeis.org

6, 24, 30, 54, 24, 84, 36, 60, 66, 42, 96, 84, 126, 90, 150, 84, 120, 36, 204, 210, 210, 60, 216, 132, 96, 336, 72, 144, 240, 294, 84, 252, 360, 114, 156, 180, 210, 120, 210, 420, 168, 270, 264, 168, 384, 240, 468, 126, 180, 336, 336, 504, 264, 330, 486, 216

Offset: 0

Eric W. Weisstein

nonn

This is the ordering of triangles used for A316841.

Eric Weisstein's World of Mathematics, Heronian Triangle.

The sides are given by A046128, A046129, A046130.
Cf. A055595, A072294, A316841, A316853.
Range of values: A383413.

sideMax = 60; r[c_] := Reap[Do[ p = (a + b + c)/2; red = Reduce[ area > 1 && a < b < c && area^2 == p*(p - a)*(p - b)*(p - c), area, Integers]; If[red =!= False, sol = {a, b, c, area} /. {ToRules[red]}; Sow[sol]], {b, 1, c - 1}, {a, c - b, b - 1}]]; triangles = Flatten[ Reap[ Do[rc = r[c]; If[rc[[2]] =!= {}, Sow[rc[[2, 1]]]], {c, 5, sideMax}]][[2, 1]] , 2]; Sort[ triangles, Which[#1[[3]] < #2[[3]], True, #1[[3]] > #2[[3]], False, #1[[2]] < #2[[2]], True,  #1[[2]] > #2[[2]], False, #1[[1]] <= #2[[1]], True, True, False] &][[All, 4]] (* Jean-François Alcover, Oct 29 2012 *)

A120131 Longest side of primitive Heronian triangles, sorted.

Original entry on oeis.org

5, 6, 8, 13, 13, 15, 15, 17, 17, 17, 20, 20, 21, 21, 24, 25, 25, 25, 26, 26, 28, 29, 29, 30, 30, 30, 35, 35, 36, 37, 37, 37, 37, 37, 39, 39, 39, 39, 40, 40, 40, 41, 41, 41, 41, 42, 44, 44, 45, 48, 48, 50, 50, 51, 51, 51, 51, 52, 52, 52, 52, 52, 52, 53, 53, 53, 53, 55, 55, 56

Offset: 1

Lekraj Beedassy, Jun 10 2006

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000
Michael Somos, Heronian Triangle Table
P. Yiu, Heron triangles with sides < 100, Recreational Mathematics, Appendix Chap. 9.3 pp. 81/360. (This is a download of 360 pages.)

Cf. A055592, A120132, A120133, A072294, A306626.

hQ[a_,b_,c_] := IntegerQ@ Sqrt@ Block[{s = (a+b+c)/2}, s (s-a) (s-b) (s-c)]; Reap[Do[If[ GCD[a, b, c] == 1 && hQ[a, b, c], Sow@ a], {a, 60}, {b, a}, {c, a-b+1, b}]][[2, 1]] (* Giovanni Resta, May 21 2016 *)

A120132 Middle side of primitive Heronian triangles, sorted on longest side (A120131), then on middle side.

Original entry on oeis.org

4, 5, 5, 12, 13, 13, 14, 10, 15, 17, 13, 15, 17, 20, 13, 17, 24, 25, 25, 25, 25, 21, 25, 17, 25, 29, 29, 34, 29, 20, 26, 30, 35, 37, 25, 28, 34, 35, 29, 37, 39, 28, 40, 40, 41, 29, 37, 39, 40, 25, 35, 41, 41, 35, 37, 38, 40, 29, 33, 39, 41, 51, 51, 35, 45, 51, 52, 51, 53, 39

Offset: 1

Lekraj Beedassy, Jun 10 2006

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000
Michael Somos, Heronian Triangle Table
P. Yiu, Heron triangles with sides < 300, Recreational Mathematics, Appendix Chap. 9.3 pp. 81/360. (This is a download of 360 pages.)

Cf. A120131, A120133, A072294.

hQ[a_,b_,c_] := IntegerQ@ Sqrt@ Block[{s = (a+b+c)/2}, s (s-a) (s-b) (s-c)]; Reap[Do[If[ GCD[a, b, c] == 1 && hQ[a, b, c], Sow@ b], {a, 60}, {b, a}, {c, a-b+1, b}]][[2, 1]] (* Giovanni Resta, May 21 2016 *)

A120133 Shortest side of primitive Heronian triangles, sorted on longest side(A120131), then on middle side(A120132) and finally on shortest side.

Original entry on oeis.org

3, 5, 5, 5, 10, 4, 13, 9, 8, 16, 11, 7, 10, 13, 13, 12, 7, 14, 3, 17, 17, 20, 6, 17, 11, 5, 8, 15, 25, 19, 15, 13, 12, 24, 16, 17, 25, 10, 29, 13, 25, 15, 9, 17, 18, 29, 15, 17, 13, 25, 29, 21, 39, 26, 20, 25, 13, 27, 25, 37, 15, 5, 25, 24, 28, 4, 51, 26, 20, 25, 53, 33, 41, 17, 15, 11

Offset: 1

Lekraj Beedassy, Jun 10 2006

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000
Michael Somos, Heronian Triangle Table
P. Yiu, Heron triangles with sides < 100, Recreational Mathematics, Appendix Chap. 9.3 pp. 81/360. (This is a download of 360 pages.)

Cf. A120131, A120132, A072294.

hQ[a_,b_,c_] := IntegerQ@ Sqrt@ Block[{s = (a+b+c)/2}, s (s-a) (s-b) (s-c)]; Reap[Do[If[ GCD[a, b, c] == 1 && hQ[a, b, c], Sow@c], {a, 60}, {b, a}, {c, a-b+1, b}]][[2, 1]] (* Giovanni Resta, May 21 2016 *)

A224301 Sorted areas of primitive integer Heronian triangles.

Original entry on oeis.org

6, 12, 12, 24, 30, 36, 36, 42, 60, 60, 60, 60, 66, 72, 84, 84, 84, 84, 90, 90, 114, 120, 120, 120, 126, 126, 126, 132, 156, 156, 168, 168, 168, 180, 180, 198, 204, 210, 210, 210, 210, 210, 210, 216, 234, 240, 252, 252, 252, 264, 264, 270, 288, 300, 300, 306

Offset: 1

Mihir Mathur, Apr 03 2013

nonn

The sequence gives the sorted areas of primitive triangles which have integer side lengths and integer areas.

			The smallest Heronian triangle is (3,4,5) as perimeter and area are integers. The first term of the sequence is thus the area of this triangle, which is 6.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Michael Somos, Heronian Triangle Table

Cf. A120131, A120132, A120133, A072294, A188158.

AMax=400;
Do[
  c=p/b;
  a1=Sqrt[b^2+c^2+2Sqrt[b^2c^2-4A^2]];
  a2=Sqrt[b^2+c^2-2Sqrt[b^2c^2-4A^2]];
  If[IntegerQ[a2]&&GCD[a2,b,c]==1&&a1>a2>=b,A//Sow(*{A,a2,b,c}//Sow*)];
  If[IntegerQ[a1]&&GCD[a1,b,c]==1,A//Sow(*{A,a1,b,c}//Sow*)];
  ,{A,6,AMax,6}
  ,{p,4A^2//Divisors//Select[#,EvenQ[#]&&#>=2A&]&//#/2+2A^2/#&//Select[#,IntegerQ]&}
  ,{b,p//Divisors//Select[#,#^2>=p&]&}
]//Reap//Last//Last
{a1,a2,c}=.;
(* Albert Lau, May 20 2016 *)

Definition corrected by Giovanni Resta, Apr 03 2013

More terms from Giovanni Resta, Apr 03 2013

cp's OEIS Frontend

A083875 One-sixth the areas of primitive Heronian triangles sorted.

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A046131 Areas of scalene integer Heronian triangles (A046128, A046129, A046130) sorted by increasing c and b.

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Mathematica

A120131 Longest side of primitive Heronian triangles, sorted.

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Mathematica

A120132 Middle side of primitive Heronian triangles, sorted on longest side (A120131), then on middle side.

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Mathematica

A120133 Shortest side of primitive Heronian triangles, sorted on longest side(A120131), then on middle side(A120132) and finally on shortest side.

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Mathematica

A224301 Sorted areas of primitive integer Heronian triangles.

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