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A155946 Numbers d for which the volume of the regular d-dimensional simplex of unit edge is rational. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 3 15:34 EST 2021. Contains 349463 sequences. (Running on oeis4.)