cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Lekraj Beedassy, Jun 10 2006

Keywords

Links

Crossrefs

Cf. A120131, A120132, A072294.

Programs

  • Mathematica
    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 *)

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

Views

Author

Lekraj Beedassy, Jun 10 2006

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

A072294 Areas of primitive Heronian triangles sorted by longest side, then by middle side and finally shortest side.

Original entry on oeis.org

6, 12, 12, 30, 60, 24, 84, 36, 60, 120, 66, 42, 84, 126, 60, 90, 84, 168, 36, 204, 210, 210, 60, 120, 132, 72, 84, 252, 360, 114, 156, 180, 210, 420, 120, 210, 420, 168, 420, 240, 468, 126, 180, 336, 360, 420, 264, 330, 252, 168, 504, 420, 780, 420, 306, 456, 156
Offset: 1

Views

Author

Lekraj Beedassy, Jul 12 2002

Keywords

Links

Crossrefs

Cf. A120131, A120132, A120133, A055595.

Programs

  • Mathematica
    nn = 200; 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}]; lst (* T. D. Noe, Mar 23 2011 *)

Extensions

More terms from Ray Chandler, Jul 02 2004

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

Views

Author

Mihir Mathur, Apr 03 2013

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

Definition corrected by Giovanni Resta, Apr 03 2013
More terms from Giovanni Resta, Apr 03 2013

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

Views

Author

Mihir Mathur, Apr 04 2013

Keywords

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

Examples

			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.

References

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

Links

Crossrefs

Cf. A120131, A120132, A120133, A224301.

Programs

  • 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 *)

Extensions

Corrected and extended by Giovanni Resta, Apr 04 2013

A272388 Longest side of Heronian tetrahedron.

Original entry on oeis.org

117, 160, 203, 225, 234, 318, 319, 319, 320, 351, 406, 429, 450, 468, 468, 480, 585, 595, 595, 595, 609, 612, 636, 638, 638, 640, 671, 675, 680, 680, 697, 697, 702, 741, 780, 800, 812, 819, 858, 884, 884, 888, 900, 925, 935, 936, 936, 954, 957, 957, 960, 990, 990
Offset: 1

Views

Author

Albert Lau, May 19 2016

Keywords

Comments

A Heronian tetrahedron or perfect tetrahedron is a tetrahedron whose edge lengths, face areas and volume are all integers.

Examples

			The following are examples of Heronian tetrahedra.
dAB, dAC, dBC, dCD, dBD, dAD, SABC,  SABD,  SACD,  SBCD,  Volume
117, 84,  51,  52,  53,  80,  1890,  1800,  2016,  1170,  18144
160, 153, 25,  39,  56,  120, 1872,  2688,  1404,  420,   8064
203, 195, 148, 203, 195, 148, 13650, 13650, 13650, 13650, 611520
225, 200, 65,  119, 156, 87,  6300,  4914,  2436,  3570,  35280
234, 168, 102, 104, 106, 160, 7560,  7200,  8064,  4680,  145152
318, 221, 203, 42,  175, 221, 22260, 18564, 4620,  2940,  206976
319, 318, 175, 175, 210, 221, 26796, 23100, 18564, 14700, 1034880
319, 318, 175, 203, 252, 221, 26796, 27720, 22260, 17640, 1241856
320, 306, 50,  78,  112, 240, 7488,  10752, 5616,  1680,  64512
351, 252, 153, 156, 159, 240, 17010, 16200, 18144, 10530, 489888
where
dPQ is the distance between vertices P and Q and
SPQR is the area of triangle PQR.

Links

Crossrefs

Cf. A120131, A120132, A120133.

Programs

  • Mathematica
    aMax=360(*WARNING:takes a long time*);
heron=1/4Sqrt[(#1+#2+#3)(-#1+#2+#3)(#1-#2+#3)(#1+#2-#3)]&;
cayley=1/24Sqrt[2Det[{
  {0,1,1,1,1},
  {1,0,#1^2,#2^2,#6^2},
  {1,#1^2,0,#3^2,#5^2},
  {1,#2^2,#3^2,0,#4^2},
  {1,#6^2,#5^2,#4^2,0}
}]]&;
Do[
  S1=heron[a,b,c];
  If[S1//IntegerQ//Not,Continue[]];
  Do[
    S2=heron[a,e,f];
    If[S2//IntegerQ//Not,Continue[]];
    Do[
      If[b==e&&c>f||b==f&&c>e,Continue[]];
      S3=heron[b,d,f];
      If[S3//IntegerQ//Not,Continue[]];
      S4=heron[c,d,e];
      If[S4//IntegerQ//Not,Continue[]];
      V=cayley[a,b,c,d,e,f];
      If[V//IntegerQ//Not,Continue[]];
      If[V==0,Continue[]];
      a//Sow(*{a,b,c,d,e,f,S1,S2,S3,S4,V}//Sow*);
    ,{d,Sqrt[((b^2-c^2+e^2-f^2)/(2a))^2+4((S1-S2)/a)^2]//Ceiling,Min[a,Sqrt[((b^2-c^2+e^2-f^2)/(2a))^2+4((S1+S2)/a)^2]]}];
  ,{e,a-b+1,b},{f,a-e+1,b}];
,{a,117,aMax},{b,a/2//Ceiling,a},{c,a-b+1,b}]//Reap//Last//Last

Extensions

a(11)-a(53) from Giovanni Resta, May 20 2016
Showing 1-6 of 6 results.

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