login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A272390 Longest side of primitive Heronian tetrahedron with 4 congruent triangle faces. 0
203, 888, 1804, 2431, 2873 (list; graph; refs; listen; history; text; internal format)
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 4 edge lengths share no common factor.

Properties:

1. 3 pairs of opposite edge lengths are equal.

2. The perimeter must be an even number.

3. The faces are acute triangles, and cannot be isosceles triangle.

It is known that 5512,8484,11275,19695,32708,294175,683787 are in the sequence.

LINKS

Table of n, a(n) for n=1..5.

EXAMPLE

Below shows some example: (might contains gap)

a,     b,     c,     S,         V

203,   195,   148,   13650,     611520

888,   875,   533,   223860,    37608480

1804,  1479,  1183,  870870,    214582368

2431,  2296,  2175,  2277660,   1403038560

2873,  2748,  1825,  2419950,   1355172000

5512,  5215,  1887,  4919460,   1377448800

8484,  6625,  6409,  20980050,  30546952800

11275, 10136, 8619,  41861820,  103147524480

19695, 16448, 13073, 106675680, 323290060800

32708, 31493, 24525, 363332970, 2685757314240

MATHEMATICA

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}

}]]&;

aMin=203;

aMax=2000(*WARNING:runs very slow*);

Do[

  If[GCD[a, b, c]>1, Continue[]];

  S=heron[a, b, c];

  If[S//IntegerQ//Not, Continue[]];

  V=cayley[a, b, c, a, b, c];

  If[V//IntegerQ//Not, Continue[]];

  a(*{a, b, c, S, V}*)//Sow;

  , {a, aMin, aMax}

  , {b, a/Sqrt[2]//Ceiling, a-1}

  , {c, Mod[a+b, 2, Floor[Sqrt[a^2-b^2]]+1], b-1, 2}

]//Reap//Last//Last(*//TableForm*)

{S, V}=.;

(*

(*this piece of code runs much faster but might contains gap*)

mMax=100;

Do[

  {a, b, c}={n(m^2+k^2), m(n^2+k^2), (m+n)(m n-k^2)};

  {a, b, c}={a, b, c}/GCD[a, b, c];

  V=cayley[a, b, c, a, b, c];

  If[V//IntegerQ//Not, Continue[]];

  a(*{a, b, c, heron[a, b, c], V}*)//Sow

  , {m, mMax}

  , {n, m-1}

  , {k, Floor[Sqrt[(m^2 n)/(2m+n)]+1], n-1}

]//Reap//Last//Last//Union(*TableForm*)

{a, b, c, V}=.;

*)

CROSSREFS

Sequence in context: A240903 A250751 A211565 * A320557 A270770 A294056

Adjacent sequences:  A272387 A272388 A272389 * A272391 A272392 A272393

KEYWORD

nonn,more

AUTHOR

Albert Lau, May 26 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 19 11:09 EST 2019. Contains 329319 sequences. (Running on oeis4.)