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%I #13 Jun 14 2020 15:37:03
%S 2,4,5,6,10,12,13,14,16,18,20,21,22,26,28,29,30,32,34,36,37,38,40,41,
%T 42,44,45,46,50,52,53,54,56,58,60,61,62,64,65,66,68,69,70,72,74,76,77,
%U 78,82,84,85,86,88,90,92,93,94,96,98,100,101,102,104,106,108
%N Dimensions d such that the integer lattice Z^d does not contain the vertices of a regular d-simplex.
%C List contains d such that (1) d is even and d+1 is not a square, or (2) d == 1 (mod 4) and d+1 is not a sum of two squares; proved by Schoenberg.
%H Hiroshi Maehara and Horst Martini, <a href="https://doi.org/10.1007/s00010-018-0557-4">Elementary geometry on the integer lattice</a>, Aequationes mathematicae, 92 (2018), 763-800. See Sec. 3.2.
%H I. J. Schoenberg, <a href="https://doi.org/10.1112/jlms/s1-12.45.48">Regular Simplices and Quadratic Forms</a>, J. London Math. Soc. 12 (1937) 48-55.
%e 2 is in the list because there is no equilateral triangle in the plane whose vertices all have integer coordinates.
%e 3 is not in the list because there is a regular tetrahedron in space whose vertices have integer coordinates; e.g. (1,1,0), (1,0,1), (0,1,1), (0,0,0).
%Y Complement of A096315.
%Y Cf. A000290, A001481, A022544, A097269.
%K nonn
%O 1,1
%A _Harry Richman_, May 08 2020
%E More terms from _Jinyuan Wang_, May 09 2020