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

Author

Hugo Pfoertner, Mar 19 2024

Keywords

• nonn
• look

Examples

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

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.

Crossrefs

Cf. A097125, A371072, A371345.

Subset of A371071.

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

Programs

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]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]= Vmin || Vmin=CM); Vmin/2} \\ M. F. Hasler, Dec 02 2024

Extensions

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