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A096315 Numbers n such that the n-dimensional integer lattice Z^n contains n+1 equidistant points (i.e., the vertices of a regular n-simplex). 1
1, 3, 7, 8, 9, 11, 15, 17, 19, 23, 24, 25, 27, 31, 33, 35, 39, 43, 47, 48, 49, 51, 55, 57, 59, 63, 67, 71, 73, 75, 79, 80, 81, 83, 87, 89, 91, 95, 97, 99, 103, 105, 107, 111, 115, 119, 120, 121, 123, 127, 129, 131, 135, 139, 143, 145, 147, 151, 155, 159, 161, 163, 167 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Schoenberg proved that a regular n-simplex can be inscribed in Z^n in the following cases and no others: (1) n is even and n+1 is a square; (2) n == 3 (mod 4); (3) n == 1 (mod 4) and n+1 is the sum of two squares.

LINKS

Table of n, a(n) for n=1..63.

I. J. Schoenberg, Regular Simplices and Quadratic Forms, J. London Math. Soc. 12 (1937) 48-55.

EXAMPLE

There is no equilateral triangle in the plane whose vertices have integer coordinates, so 2 is not on the list. But there is a regular tetrahedron in space whose vertices have integer coordinates, namely (0,0,0), (0,1,1), (1,0,1), (1,1,0), hence 3 is on the list.

MAPLE

select(n->(is(n, even) and issqr(n+1)) or (n mod 4 = 3) or ((n mod 4 = 1) and (numtheory[sum2sqr](n+1)<>[])), [ $1..200]);

CROSSREFS

Sequence in context: A122987 A047530 A265350 * A286395 A112680 A096079

Adjacent sequences:  A096312 A096313 A096314 * A096316 A096317 A096318

KEYWORD

easy,nonn

AUTHOR

David Radcliffe, Aug 01 2004

STATUS

approved

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Last modified January 17 09:54 EST 2020. Contains 330949 sequences. (Running on oeis4.)