A272390 - OEIS

%I #13 May 28 2016 07:26:11

%S 203,888,1804,2431,2873

%N Longest side of primitive Heronian tetrahedron with 4 congruent triangle faces.

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

%C Primitive tetrahedron means 4 edge lengths share no common factor.

%C Properties:

%C 1. 3 pairs of opposite edge lengths are equal.

%C 2. The perimeter must be an even number.

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

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

%e Below shows some example: (might contains gap)

%e a,     b,     c,     S,         V

%e 203,   195,   148,   13650,     611520

%e 888,   875,   533,   223860,    37608480

%e 1804,  1479,  1183,  870870,    214582368

%e 2431,  2296,  2175,  2277660,   1403038560

%e 2873,  2748,  1825,  2419950,   1355172000

%e 5512,  5215,  1887,  4919460,   1377448800

%e 8484,  6625,  6409,  20980050,  30546952800

%e 11275, 10136, 8619,  41861820,  103147524480

%e 19695, 16448, 13073, 106675680, 323290060800

%e 32708, 31493, 24525, 363332970, 2685757314240

%t heron=1/4Sqrt[(#1+#2+#3)(-#1+#2+#3)(#1-#2+#3)(#1+#2-#3)]&;

%t cayley=1/24Sqrt[2Det[{

%t   {0,1,1,1,1},

%t   {1,0,#1^2,#2^2,#6^2},

%t   {1,#1^2,0,#3^2,#5^2},

%t   {1,#2^2,#3^2,0,#4^2},

%t   {1,#6^2,#5^2,#4^2,0}

%t }]]&;

%t aMin=203;

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

%t Do[

%t   If[GCD[a,b,c]>1,Continue[]];

%t   S=heron[a,b,c];

%t   If[S//IntegerQ//Not,Continue[]];

%t   V=cayley[a,b,c,a,b,c];

%t   If[V//IntegerQ//Not,Continue[]];

%t   a(*{a,b,c,S,V}*)//Sow;

%t   ,{a,aMin,aMax}

%t   ,{b,a/Sqrt[2]//Ceiling,a-1}

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

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

%t {S,V}=.;

%t (*

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

%t mMax=100;

%t Do[

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

%t   {a,b,c}={a,b,c}/GCD[a,b,c];

%t   V=cayley[a,b,c,a,b,c];

%t   If[V//IntegerQ//Not,Continue[]];

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

%t   ,{m,mMax}

%t   ,{n,m-1}

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

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

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

%t *)

%K nonn,more

%O 1,1

%A _Albert Lau_, May 26 2016