A232209 - OEIS

%I #42 Aug 05 2021 21:45:27

%S 1,2,16,48,10140,6552,721125376,3620732928,278799279816,29925033224,

%T 229926056690973293936640,892398340719534485274624,

%U 603207249820766251389767637583758341569376980491272,240171846906336440253785749946778562802349467993472

%N Height of algebraic number 1 + sqrt(2) + ... + sqrt(n).

%C Let p(z) be the monic minimal polynomial of sum(j=1..n, sqrt(j)) over the integers.

%C a(n) is the maximum of the absolute values of the coefficients of p(z).

%C The degree of p(z) is at most A048656(n). Is it always equal to A048656(n)?

%H Robert Israel, <a href="/A232209/b232209.txt">Table of n, a(n) for n = 1..18</a>

%H Springer, Encyclopedia of Mathematics, <a href="http://www.encyclopediaofmath.org/index.php/Algebraic_number">Algebraic number</a>

%e For n = 1 the minimal polynomial of 1 is z - 1 so a(1) = 1.

%e For n = 2 the minimal polynomial of 1 + sqrt(2) is z^2 - 2*z - 1 so a(2) = 2.

%p for n from 1 to 15 do

%p    a:= convert(add(sqrt(i),i=1..n), RootOf);

%p    P:= evala(Norm(a-z));

%p    A[n]:= max(map(abs,[coeffs(P,z)]));

%p od:

%p seq(A[n],n=1..15); # _Robert Israel_, Sep 10 2014

%t a[n_] := CoefficientList[ MinimalPolynomial[ Sqrt[Range[n]] // Total, x], x] // Abs // Max; Array[a, 12] (* _Jean-François Alcover_, Apr 29 2019 *)

%o (GAP) a:=function(n)

%o return MinimalPolynomial(Rationals, Sum([1..n], x->Sqrt(x)));

%o end; # _Charles R Greathouse IV_, Sep 12 2014

%Y Cf. A048656.

%K nonn

%O 1,2

%A _Robert Israel_, Sep 10 2014