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

2, 11, 26, 47, 54, 107, 146, 191, 242, 299, 191, 134, 146, 146, 151, 767, 423, 151, 854, 558, 764, 491, 503, 464, 146, 146, 431, 944, 666, 146, 146, 350, 599, 311, 599, 511, 1719, 2286, 944, 1871, 1679, 990, 2714, 1907, 990, 551, 959, 1199, 1244, 990, 1206, 854, 764

Offset

1,1

Links

Hugo Pfoertner, Table of n, a(n) for n = 1..450

IBM Research, Tetrahedron Volumes, Ponder This Challenge November 2024.

Sascha Kurz, Enumeration of integral tetrahedra, arXiv:0804.1310 [math.CO], 2008.

Hugo Pfoertner, Plot of log_10(n) vs n, using Plot 2.

Example

a(1) = 2 corresponds to the regular tetrahedron with all edges equal to 1. Its volume is sqrt(2/144) = 0.11785113...

Prog

(PARI) \\ See A371345. Replace final #Set(Vec(L)) by vecmin(Vec(L))/2
\\ Second version using simple minded loops and triangle inequalities
\\ Not suitable for larger n
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;
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
(PARI) /* equivalent to the preceding, but simplified */
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

Crossrefs

Cf. A097125, A371072, A371345.

Subset of A371071.

A001014(n)/72 are the corresponding maximum squared volumes.

Sequence in context: A296285 A077482 A141428 * A104085 A080663 A248118

Adjacent sequences: A371341 A371342 A371343 * A371345 A371346 A371347

Keyword

nonn,look

Author

Hugo Pfoertner, Mar 19 2024

Extensions

a(33), a(37), a(38), and a(43) corrected by Hugo Pfoertner, Dec 03 2024

Status

approved