

A222624


Sequence of dimensions of a regular simplex with unit edge that has at least one interhypersphere with a rational radius (including the radii of the inhypersphere and circumhypersphere).


0



1, 7, 8, 17, 23, 31, 35, 49, 55, 62, 71, 74, 80, 89, 95, 97, 119, 120, 127, 143, 146, 161, 174, 199, 215, 223, 224, 241, 244, 251, 287, 288, 293, 299, 305, 323, 337, 359, 360, 362, 383, 391, 424, 440, 449, 479, 483, 489, 503, 511, 566, 574, 575, 577, 587, 599
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OFFSET

1,2


COMMENTS

The radius of the interhypersphere of a regular simplex of dimension k with unit edge is given by r=Sqrt((kj)/(2*(k+1)*(j+1))) where 0<=j<k and j is the dimension of the hypersurface that the interhypersphere is tangent to. j=0 gives the radius of the circumhypersphere and j=k1 gives the radius of the inhypersphere. The sequence identifies those dimensions k for which at least one interhypersphere has a rational radius.


LINKS

Table of n, a(n) for n=1..56.
Wikipedia, 8simplex.


FORMULA

a(n)=k is the nth occurrence for which k=sqrt((kj)/(2*(k+1)*(j+1))) is rational for at least one value of j in the range 0<=j<k.


EXAMPLE

a(3)=8 because it is the 3rd value of k with k=8 that has an interhypersphere of rational radius. In fact in dimension 8 the regular simplex with unit edge has 4 interhyperspheres of rational radius. They are 2/3 at j=0 (circumhypersphere), 1/3 at j=2, 1/6 at j=5 and 1/12 at j=7 (inhypersphere).


MATHEMATICA

maxlen = 100; getlist[n_] := (d = 0; sublist = {}; While[d < n, If[IntegerQ[Numerator[Sqrt[(nd)/(2(n+1)(d+1))]]*Denominator[Sqrt[(nd)/(2(n+1)(d+1))]]], (sublist=Append[sublist, d]; d++), d++]]; sublist); m = 1; fulllist = {}; While[Length[fulllist] <= maxlen, If[Length[getlist[m]]==0, m++, (fulllist=Append[fulllist, m]; m++)]; ]; fulllist


CROSSREFS

Cf. A155946.
Sequence in context: A054312 A326658 A106678 * A214788 A155946 A144614
Adjacent sequences: A222621 A222622 A222623 * A222625 A222626 A222627


KEYWORD

nonn


AUTHOR

Frank M Jackson, Feb 26 2013


STATUS

approved



