A232209 Height of algebraic number 1 + sqrt(2) + ... + sqrt(n).

1, 2, 16, 48, 10140, 6552, 721125376, 3620732928, 278799279816, 29925033224, 229926056690973293936640, 892398340719534485274624, 603207249820766251389767637583758341569376980491272, 240171846906336440253785749946778562802349467993472

Offset

1,2

Comments

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

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

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

Links

Robert Israel, Table of n, a(n) for n = 1..18

Springer, Encyclopedia of Mathematics, Algebraic number

Example

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

Maple

for n from 1 to 15 do
   a:= convert(add(sqrt(i), i=1..n), RootOf);
   P:= evala(Norm(a-z));
   A[n]:= max(map(abs, [coeffs(P, z)]));
od:
seq(A[n], n=1..15); # Robert Israel, Sep 10 2014

Mathematica

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

Prog

(GAP) a:=function(n)
return MinimalPolynomial(Rationals, Sum([1..n], x->Sqrt(x)));
end; # Charles R Greathouse IV, Sep 12 2014

Crossrefs

Cf. A048656.

Sequence in context: A159010 A223219 A063721 * A012180 A058376 A295906

Adjacent sequences: A232206 A232207 A232208 * A232210 A232211 A232212

Keyword

nonn

Author

Robert Israel, Sep 10 2014

Status

approved