A125866 Odd numbers k such that cos(2*Pi/k) is an algebraic number of a 3-smooth degree, but not 2-smooth.

7, 9, 13, 19, 21, 27, 35, 37, 39, 45, 57, 63, 65, 73, 81, 91, 95, 97, 105, 109, 111, 117, 119, 133, 135, 153, 163, 171, 185, 189, 193, 195, 219, 221, 243, 247, 259, 273, 285, 291, 315, 323, 327, 333, 351, 357, 365, 399, 405, 433, 455, 459, 481, 485, 487, 489

Offset: 1

Author

Artur Jasinski, Dec 13 2006

Keywords

• nonn

Comments

Odd terms of A051913.

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

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Maple

filter:= proc(n) local r,a,b;
  r:= numtheory:-phi(n);
  a:= padic:-ordp(r,2);
  b:= padic:-ordp(r,3);
  if b = 0 then return false fi;
  r = 2^a*3^b;
end proc:
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, May 11 2020

Mathematica

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

Extensions

Edited by Don Reble, Apr 24 2007