A272388 Longest side of Heronian tetrahedron.

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

1,1

Comments

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

Links

Table of n, a(n) for n=1..53.

R. H. Buchholz, Perfect Pyramids, Bull. Austral. Math. Soc. 45, 353-368, 1992.

Susan H. Marshall and Alexander R. Perlis, Heronian Tetrahedra Are Lattice Tetrahedra, American Mathematical Monthly 120:2 (2013), 140-149.

Ivars Peterson, Perfect pyramids.

Eric Weisstein's World of Mathematics, Heronian Tetrahedron.

Example

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.

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

Crossrefs

Cf. A120131, A120132, A120133.

Sequence in context: A015706 A095625 A109023 * A272389 A201021 A112877

Adjacent sequences: A272385 A272386 A272387 * A272389 A272390 A272391

Keyword

nonn

Author

Albert Lau, May 19 2016

Extensions

a(11)-a(53) from Giovanni Resta, May 20 2016

Status

approved