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A155946 Numbers d for which the volume of the regular d-dimensional simplex of unit edge is rational. 2

%I

%S 0,1,7,8,17,24,31,48,49,71,80,97,120,127,161,168,199,224,241,287,288,

%T 337,360,391,440,449,511,528,577,624,647,721,728,799,840,881,960,967

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

%H Charles R Greathouse IV, <a href="/A155946/b155946.txt">Table of n, a(n) for n = 1..10000</a>

%F 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.

%t getrat[n_] := Sqrt[(n+1)/2^n];

%t nextdim[m_] := (p=m+1;While[!IntegerQ[Numerator[getrat[p]]*Denominator[getrat[p]]], p++]; p);

%t Table[Nest[nextdim, -1, q], {q, 1, 100}] (* _Frank M Jackson_, Feb 26 2013 *)

%o (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

%o (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

%K nonn

%O 1,3

%A _David Pasino_, Jan 31 2009

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Last modified January 18 09:26 EST 2022. Contains 350454 sequences. (Running on oeis4.)