A272389 Longest side of primitive Heronian tetrahedron.

117, 160, 203, 225, 318, 319, 319

Offset

1,1

Comments

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

Primitive tetrahedron means 6 sides don't share a common factor.

Links

Table of n, a(n) for n=1..7.

Example

see A272388

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[GCD[a, b, c, d, e, f] > 1, Continue[]];
      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. A272388

Sequence in context: A095625 A109023 A272388 * A201021 A112877 A064180

Adjacent sequences: A272386 A272387 A272388 * A272390 A272391 A272392

Keyword

nonn,more

Author

Albert Lau, May 21 2016

Status

approved