A224301 -id:A224301 - cp's OEIS frontend

A227166 Areas of indecomposable non-Pythagorean primitive integer Heronian triangles, sorted increasingly.

72, 126, 168, 252, 252, 288, 336, 336, 396, 396, 420, 420, 420, 420, 456, 462, 528, 528, 624, 714, 720, 720, 756, 792, 798, 840, 840, 840, 840, 864, 924, 924, 924, 924, 936, 990, 1008, 1092, 1092, 1188, 1200, 1218, 1248, 1260, 1260, 1320, 1320, 1320

Offset: 1

Frank M Jackson, Jul 03 2013

nonn

An indecomposable integer Heronian triangle that is not Pythagorean cannot be decomposed into two separate Pythagorean triangles because it has no integer altitudes.

See comments in A227003 about the Mathematica program below to ensure that all primitive Heronian areas up to 1320 are captured.

			a(2) = 126 as this is the second smallest area of an indecomposable non-Pythagorean primitive Heronian triangle. The triple is (5,51,52).

Paul Yiu, Heron triangles which cannot be decomposed into two integer right triangles, 2008

Cf. A083875, A224301, A227003, A239978.

nn=1320; 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]] && !IntegerQ[2Sqrt[area2]/a] && !IntegerQ[2Sqrt[area2]/b] && !IntegerQ[2Sqrt[area2]/c], AppendTo[lst, Sqrt[area2]]]], {a, 3, nn}, {b, a}, {c, b}]; Sort@Select[lst, #<=nn &] (* using T. D. Noe's program A083875 *)

Name clarified by Frank M Jackson, Mar 17 2014

A239978 Areas of indecomposable primitive integer Heronian triangles (including primitive Pythagorean triangles), in increasing order.

Original entry on oeis.org

6, 30, 60, 72, 84, 126, 168, 180, 210, 210, 252, 252, 288, 330, 336, 336, 396, 396, 420, 420, 420, 420, 456, 462, 504, 528, 528, 546, 624, 630, 714, 720, 720, 756, 792, 798, 840, 840, 840, 840, 840, 864, 924, 924, 924, 924, 924, 936, 990, 990, 1008

Offset: 1

Frank M Jackson, Mar 30 2014

nonn

An indecomposable Heronian triangle is a Heronian triangle that cannot be split into two Pythagorean triangles. In other words, it has no integer altitude that is not a side of the triangle. Note that all primitive Pythagorean triangles are indecomposable.

See comments in A227003 about the Mathematica program below to ensure that all primitive Heronian areas up to 1008 are captured.

			a(5) = 84 as this is the fifth ordered area of an indecomposable primitive Heronian triangle. The triple is (7,24,25) and it is Pythagorean.

Paul Yiu, Heron triangles which cannot be decomposed into two integer right triangles, 2008.

Cf. A083875, A224301, A227003, A227166.

nn=1008; 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]]&&((!IntegerQ[2Sqrt[area2]/a]&&!IntegerQ[2Sqrt[area2]/b]&&!IntegerQ[2Sqrt[area2]/c])||(c^2+b^2==a^2)), AppendTo[lst, Sqrt[area2]]]], {a,3,nn}, {b,a}, {c,b}]; Sort@Select[lst, #<=nn &] (*using T. D. Noe's program A083875*)

A224302 Sorted perimeters of primitive Heronian triangles.

Original entry on oeis.org

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

Offset: 1

Mihir Mathur, Apr 04 2013

nonn

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

			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.

L. E. Dickson, History of the Theory of Numbers, vol. II: Diophantine Analysis, Dover, 2005, p. 196. [21a]

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

Cf. A120131, A120132, A120133, A224301.

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 *)

Corrected and extended by Giovanni Resta, Apr 04 2013

A247381 The area of a primitive Heronian triangle K, such that K = k^2*n for the least k, where n is the sequence index.

Original entry on oeis.org

36, 72, 12, 36, 180, 6, 252, 72, 36, 90, 396, 12, 468, 126, 60, 7056, 2448, 72, 684, 180, 84, 198, 20700, 24, 900, 234, 5292, 252, 4176, 30, 1116, 288, 132, 306, 1260, 36, 1332, 5472, 156, 360, 5904, 42, 1548, 396, 180, 1656, 82908, 1200, 7056, 1800, 204, 468, 30528, 216

Offset: 1

Frank M Jackson, Sep 15 2014

nonn

It has been proved that every positive integer is the area of some rational sided Heronian triangle. Therefore for all positive integers n there exists a primitive Heronian triangle such that for some least k^2 its area K = k^2*n. The Mathematica program limits searches to all primitive Heronian triangles whose largest side does not exceed 1000 and returns 0 if no area is found.

			a(23)=30^2*23=20700 and the primitive Heronian triangle has sides (73, 579, 598).

N. J. Fine, On Rational Triangles, Mathematical Association of America, 83-7 (1976), 517-521.
Jaap Top and Noriko Yui, Congruent number problems and their variants, Algorithmic Number Theory, MSRI Publications Volume 44, 2008, p. 621.
Yahoo Answers, Is each positive integer the area of some triangle with rational sides?

Cf. A224301.

getarea[n0_] := (area1=0; Do[If[IntegerQ[area=Sqrt[(a+b+c)(a+b-c)(a-b+c)(-a+b+c)/16]]&&area>0&&IntegerQ[k=Sqrt[area/n0]]&&GCD[a, b, c]==1, area1=area; Break[]], {c, 3, 1000}, {b, 1, c}, {a, 1, b}]; area1); Table[getarea[n], {n, 1, 100}]

Updated and edited by Frank M Jackson, Jun 14 2016

A240240 Consider primitive Heronian triangles with integer area and with sides {m, m+1, c}, where c > m+1. The sequence gives the possible values of m.

Original entry on oeis.org

3, 9, 13, 19, 20, 33, 51, 65, 73, 99, 119, 129, 163, 170, 174, 193, 201, 203, 220, 243, 260, 269, 287, 289, 339, 362, 377, 393, 450, 451, 513, 532, 559, 579, 615, 649, 696, 702, 714, 723, 740, 771, 801, 883, 909, 940, 969, 975, 1059, 1112, 1153, 1155, 1156, 1164, 1251, 1299, 1325, 1332, 1353, 1424, 1455, 1459, 1569, 1605, 1615, 1683, 1690, 1716, 1801, 1869, 1919, 1923

Offset: 1

Zak Seidov, Apr 03 2014

nonn

Corresponding values of c are 5, 17, 15, 37, 29, 65, 101, 109, 145.
And corresponding values of area/6 are 1, 6, 14, 19, 35, 44, 85, 330, 146, 231, 1190.
The sequence includes all terms of A016064 (where c = m+2) except for the first term, 1 (case with zero area).
Note that in all cases c is odd and m+2 <= c < 2m+1.

			First triangle has sides (3,4,5) and area 6.
2nd triangle has sides (9,10,17) and area 36.
3rd triangle has sides (13,14,15) and area 84.

Cf. A083875, A016064, A224301, A227003, A239978, A227166.

re=Reap[Do[a=m;b=m+1;Do[s=(a+b+c)/2;area=Sqrt[s(s-a)(s-b)(s-c)];If[IntegerQ[area],Sow[{a,b,c,area}];Break[]],{c,2m-1,m+2,-2 }],{m,3,2000}]][[2,1]];#[[1]]&/@ re

A343769 Sorted areas of primitive Heronian triangles for which a rectangle exists with integer dimensions and with perimeter and area equal respectively to the perimeter and area of the triangle.

Original entry on oeis.org

12, 126, 624, 1260, 1800, 2100, 2850, 4536, 5292, 5580, 8820, 9900, 12600, 12642, 14850, 15600, 17640, 19110, 21756, 23400, 24948, 25200, 25536, 28350, 47040, 47304

Offset: 1

Jason Zimba, Apr 28 2021

			a(1) = 12 because 12 is the area of the 5-5-6 triangle, which is the least-area primitive Heronian triangle for which a rectangle exists with integer dimensions (2-by-6) and with perimeter (16) and area (12) equal respectively to the perimeter and area of the triangle.
a(2) = 126 because 126 is the area of the 13-20-21 triangle, which is the second-least-area primitive Heronian triangle for which a rectangle exists with integer dimensions (6-by-21) and with perimeter (54) and area (126) equal respectively to the perimeter and area of the triangle.

Jason Zimba, There are infinitely many rectangular Heronian triangles.

Subsequence of A224301.

(* Adapted from Albert Lau's program for A224301 *)
AMax = 10000;
Do[c = p/b;
    a1 = Sqrt[b^2 + c^2 + 2 Sqrt[b^2 c^2 - 4 A^2]];
    a2 = Sqrt[b^2 + c^2 - 2 Sqrt[b^2 c^2 - 4 A^2]];
    If[IntegerQ[a2] && GCD[a2, b, c] == 1 &&
      a1 > a2 >= b && (per = a2 + b + c;
       IntegerQ[(per + Sqrt[per^2 - 16 A])/4]), A // Sow(*{A,a2,b,c}//
     Sow*)];
    If[IntegerQ[a1] &&
      GCD[a1, b, c] == 1 && (per = a1 + b + c;
       IntegerQ[(per + Sqrt[per^2 - 16 A])/4]), A // Sow(*{A,a1,b,c}//
     Sow*)];, {A, 6, AMax, 6}, {p,
     4 A^2 // Divisors //
        Select[#, EvenQ[#] && # >= 2 A &] & // #/2 + 2 A^2/# & //
      Select[#, IntegerQ] &}, {b,
     p // Divisors // Select[#, #^2 >= p &] &}] // Reap // Last // Last
{a1, a2, c} =.;

cp's OEIS Frontend

A227166 Areas of indecomposable non-Pythagorean primitive integer Heronian triangles, sorted increasingly.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A239978 Areas of indecomposable primitive integer Heronian triangles (including primitive Pythagorean triangles), in increasing order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A224302 Sorted perimeters of primitive Heronian triangles.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A247381 The area of a primitive Heronian triangle K, such that K = k^2*n for the least k, where n is the sequence index.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A240240 Consider primitive Heronian triangles with integer area and with sides {m, m+1, c}, where c > m+1. The sequence gives the possible values of m.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A343769 Sorted areas of primitive Heronian triangles for which a rectangle exists with integer dimensions and with perimeter and area equal respectively to the perimeter and area of the triangle.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica