A155946 Numbers d for which the volume of the regular d-dimensional simplex of unit edge is rational.

0, 1, 7, 8, 17, 24, 31, 48, 49, 71, 80, 97, 120, 127, 161, 168, 199, 224, 241, 287, 288, 337, 360, 391, 440, 449, 511, 528, 577, 624, 647, 721, 728, 799, 840, 881, 960, 967

Offset: 1

Author

David Pasino, Jan 31 2009

Keywords

• nonn

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Programs

Mathematica

getrat[n_] := Sqrt[(n+1)/2^n];
nextdim[m_] := (p=m+1;While[!IntegerQ[Numerator[getrat[p]]*Denominator[getrat[p]]], p++]; p);
Table[Nest[nextdim, -1, q], {q, 1, 100}] (* Frank M Jackson, Feb 26 2013 *)

PARI

is(n)=if(n%2,my(o=valuation(n++,2)); o%2 && issquare(n>>o,&n) && n%2,issquare(n+1)) \\ Charles R Greathouse IV, Feb 26 2013

PARI

list(lim)=my(v=List()); forstep(q=1,sqrtint(1+lim\1), 2, listput(v,q^2-1)); for(q=1, sqrtint(1+lim\2), listput(v,2*q^2-1)); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Feb 26 2013

Formula

The volume of the regular d-dimensional simplex of unit edge is V = sqrt((d+1)/2^d)/d!. V is rational if and only if d is of the form q^2*2^k - 1 where q is odd and k is either odd or 0. The even d of this form are the odd squares minus 1. The odd d are the set generated by the function 4x + 3 from the number form 2*q^2 - 1 with q odd.