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A232209
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Height of algebraic number 1 + sqrt(2) + ... + sqrt(n).
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1
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1, 2, 16, 48, 10140, 6552, 721125376, 3620732928, 278799279816, 29925033224, 229926056690973293936640, 892398340719534485274624, 603207249820766251389767637583758341569376980491272, 240171846906336440253785749946778562802349467993472
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OFFSET
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1,2
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COMMENTS
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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)?
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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a[n_] := CoefficientList[ MinimalPolynomial[ Sqrt[Range[n]] // Total, x], x] // Abs // Max; Array[a, 12] (* Jean-François Alcover, Apr 29 2019 *)
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PROG
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(GAP) a:=function(n)
return MinimalPolynomial(Rationals, Sum([1..n], x->Sqrt(x)));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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