cp's OEIS Frontend

A272390 Longest side of primitive Heronian tetrahedron with 4 congruent triangle faces.

%I A272390 #13 May 28 2016 07:26:11
%S A272390 203,888,1804,2431,2873
%N A272390 Longest side of primitive Heronian tetrahedron with 4 congruent triangle faces.
%C A272390 A Heronian tetrahedron or perfect tetrahedron is a tetrahedron whose edge lengths, face areas and volume are all integers.
%C A272390 Primitive tetrahedron means 4 edge lengths share no common factor.
%C A272390 Properties:
%C A272390 1. 3 pairs of opposite edge lengths are equal.
%C A272390 2. The perimeter must be an even number.
%C A272390 3. The faces are acute triangles, and cannot be isosceles triangle.
%C A272390 It is known that 5512,8484,11275,19695,32708,294175,683787 are in the sequence.
%e A272390 Below shows some example: (might contains gap)
%e A272390 a,     b,     c,     S,         V
%e A272390 203,   195,   148,   13650,     611520
%e A272390 888,   875,   533,   223860,    37608480
%e A272390 1804,  1479,  1183,  870870,    214582368
%e A272390 2431,  2296,  2175,  2277660,   1403038560
%e A272390 2873,  2748,  1825,  2419950,   1355172000
%e A272390 5512,  5215,  1887,  4919460,   1377448800
%e A272390 8484,  6625,  6409,  20980050,  30546952800
%e A272390 11275, 10136, 8619,  41861820,  103147524480
%e A272390 19695, 16448, 13073, 106675680, 323290060800
%e A272390 32708, 31493, 24525, 363332970, 2685757314240
%t A272390 heron=1/4Sqrt[(#1+#2+#3)(-#1+#2+#3)(#1-#2+#3)(#1+#2-#3)]&;
%t A272390 cayley=1/24Sqrt[2Det[{
%t A272390   {0,1,1,1,1},
%t A272390   {1,0,#1^2,#2^2,#6^2},
%t A272390   {1,#1^2,0,#3^2,#5^2},
%t A272390   {1,#2^2,#3^2,0,#4^2},
%t A272390   {1,#6^2,#5^2,#4^2,0}
%t A272390 }]]&;
%t A272390 aMin=203;
%t A272390 aMax=2000(*WARNING:runs very slow*);
%t A272390 Do[
%t A272390   If[GCD[a,b,c]>1,Continue[]];
%t A272390   S=heron[a,b,c];
%t A272390   If[S//IntegerQ//Not,Continue[]];
%t A272390   V=cayley[a,b,c,a,b,c];
%t A272390   If[V//IntegerQ//Not,Continue[]];
%t A272390   a(*{a,b,c,S,V}*)//Sow;
%t A272390   ,{a,aMin,aMax}
%t A272390   ,{b,a/Sqrt[2]//Ceiling,a-1}
%t A272390   ,{c,Mod[a+b,2,Floor[Sqrt[a^2-b^2]]+1],b-1,2}
%t A272390 ]//Reap//Last//Last(*//TableForm*)
%t A272390 {S,V}=.;
%t A272390 (*
%t A272390 (*this piece of code runs much faster but might contains gap*)
%t A272390 mMax=100;
%t A272390 Do[
%t A272390   {a,b,c}={n(m^2+k^2),m(n^2+k^2),(m+n)(m n-k^2)};
%t A272390   {a,b,c}={a,b,c}/GCD[a,b,c];
%t A272390   V=cayley[a,b,c,a,b,c];
%t A272390   If[V//IntegerQ//Not,Continue[]];
%t A272390   a(*{a,b,c,heron[a,b,c],V}*)//Sow
%t A272390   ,{m,mMax}
%t A272390   ,{n,m-1}
%t A272390   ,{k,Floor[Sqrt[(m^2 n)/(2m+n)]+1],n-1}
%t A272390 ]//Reap//Last//Last//Union(*TableForm*)
%t A272390 {a,b,c,V}=.;
%t A272390 *)
%K A272390 nonn,more
%O A272390 1,1
%A A272390 _Albert Lau_, May 26 2016