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Odd numbers k such that cos(2*Pi/k) is an algebraic number of a 3-smooth degree, but not 2-smooth.
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%I #18 May 23 2022 03:55:34

%S 7,9,13,19,21,27,35,37,39,45,57,63,65,73,81,91,95,97,105,109,111,117,

%T 119,133,135,153,163,171,185,189,193,195,219,221,243,247,259,273,285,

%U 291,315,323,327,333,351,357,365,399,405,433,455,459,481,485,487,489

%N Odd numbers k such that cos(2*Pi/k) is an algebraic number of a 3-smooth degree, but not 2-smooth.

%C Odd terms of A051913.

%C This sequence is infinite (unlike A004729), because it contains any A058383(n) times any power of 3.

%C A regular polygon of a(n) sides can be constructed if one also has an angle trisector.

%H Robert Israel, <a href="/A125866/b125866.txt">Table of n, a(n) for n = 1..10000</a>

%p filter:= proc(n) local r,a,b;

%p r:= numtheory:-phi(n);

%p a:= padic:-ordp(r,2);

%p b:= padic:-ordp(r,3);

%p if b = 0 then return false fi;

%p r = 2^a*3^b;

%p end proc:

%p select(filter, [seq(i,i=3..1000,2)]); # _Robert Israel_, May 11 2020

%t Do[If[Take[FactorInteger[EulerPhi[2n+1]][[ -1]], 1]=={3},Print[2n+1]],{n,1,10000}]

%Y Cf. A004729, A051913, A058383, A125867-A125878.

%K nonn

%O 1,1

%A _Artur Jasinski_, Dec 13 2006

%E Edited by _Don Reble_, Apr 24 2007