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

1,3

Links

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

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.

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 *)

Prog

(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

Crossrefs

Sequence in context: A106678 A222624 A214788 * A144614 A308953 A143504

Adjacent sequences: A155943 A155944 A155945 * A155947 A155948 A155949

Keyword

nonn

Author

David Pasino, Jan 31 2009

Status

approved