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A371344 a(n)/144 is the minimum squared volume > 0 of a tetrahedron with integer edge lengths whose largest is n.

%I A371344 #45 Dec 10 2024 05:34:04
%S A371344 2,11,26,47,54,107,146,191,242,299,191,134,146,146,151,767,423,151,
%T A371344 854,558,764,491,503,464,146,146,431,944,666,146,146,350,599,311,599,
%U A371344 511,1719,2286,944,1871,1679,990,2714,1907,990,551,959,1199,1244,990,1206,854,764
%N A371344 a(n)/144 is the minimum squared volume > 0 of a tetrahedron with integer edge lengths whose largest is n.
%H A371344 Hugo Pfoertner, <a href="/A371344/b371344.txt">Table of n, a(n) for n = 1..450</a>
%H A371344 IBM Research, <a href="https://research.ibm.com/haifa/ponderthis/challenges/November2024.html">Tetrahedron Volumes</a>, Ponder This Challenge November 2024.
%H A371344 Sascha Kurz, <a href="https://arxiv.org/abs/0804.1310">Enumeration of integral tetrahedra</a>, arXiv:0804.1310 [math.CO], 2008.
%H A371344 Hugo Pfoertner, <a href="/plot2a?name1=A000027&amp;name2=A371344&amp;tform1=untransformed&amp;tform2=log+base+10&amp;shift=0&amp;radiop1=xy&amp;drawpoints=true">Plot of log_10(n) vs n</a>, using Plot 2.
%e A371344 a(1) = 2 corresponds to the regular tetrahedron with all edges equal to 1. Its volume is sqrt(2/144) = 0.11785113...
%o A371344 (PARI) \\ See A371345. Replace final #Set(Vec(L)) by vecmin(Vec(L))/2
%o A371344 \\ Second version using simple minded loops and triangle inequalities
%o A371344 \\ Not suitable for larger n
%o A371344 a371344(n) = {my (Vmin=oo,w=vector(6)); w[1]=n; for(w2=1,n,w[2]=w2; for(w3=1,n,w[3]=w3; for(w4=1,n,w[4]=w4; for(w5=1,n,w[5]=w5; for(w6=1,n,w[6]=w6;
%o A371344 forperm (w, v, if(v[4]+v[5]<v[6], next); if(v[4]+v[6]<v[5], next); if(v[5]+v[6]<v[4], next); if(v[1]+v[2]<v[4], next); if(v[1]+v[4]<v[2], next); if(v[2]+v[4]<v[1], next); if(v[1]+v[3]<v[5], next); if(v[1]+v[5]<v[3], next); if(v[3]+v[5]<v[1], next); if(v[2]+v[3]<v[6], next); if(v[2]+v[6]<v[3], next); if(v[3]+v[6]<v[2], next); my(CM=matdet ([0, 1, 1, 1, 1; 1, 0, v[1]^2, v[2]^2, v[3]^2; 1, v[1]^2, 0, v[4]^2, v[5]^2; 1, v[2]^2, v[4]^2, 0, v[6]^2; 1, v[3]^2, v[5]^2, v[6]^2, 0])); if (CM>0, Vmin=min(Vmin,CM)))))))); Vmin/2}; \\ return value corrected by _M. F. Hasler_, Dec 02 2024
%o A371344 (PARI) /* equivalent to the preceding, but simplified */
%o A371344 A371344(n) = {my (Vmin=oo,CM, n2=n^2); forvec(v=vector(5,k,[1,n]), v[4]+v[5]<n || v[1]+v[2]<v[4] || v[1]+v[4]<v[2] || v[2]+v[4]<v[1] || v[1]+v[3]<v[5] || v[1]+v[5]<v[3] || v[3]+v[5]<v[1] || v[2]+v[3]<n || v[2]+n<v[3] || v[3]+n<v[2] || (CM=matdet([0, 1, 1, 1, 1; 1, 0, v[1]^2, v[2]^2, v[3]^2; 1, v[1]^2, 0, v[4]^2, v[5]^2; 1, v[2]^2, v[4]^2, 0, n2; 1, v[3]^2, v[5]^2, n2, 0]))<=0 || CM >= Vmin || Vmin=CM); Vmin/2} \\ _M. F. Hasler_, Dec 02 2024
%Y A371344 Cf. A097125, A371072, A371345.
%Y A371344 Subset of A371071.
%Y A371344 A001014(n)/72 are the corresponding maximum squared volumes.
%K A371344 nonn,look
%O A371344 1,1
%A A371344 _Hugo Pfoertner_, Mar 19 2024
%E A371344 a(33), a(37), a(38), and a(43) corrected by _Hugo Pfoertner_, Dec 03 2024