[SeqFan] Algebraic degree of Cos(2Pi/n), etc.

[Antti Karttunen]

Antti Karttunen wrote:

>
> I just found this:
>
> http://www.research.att.com/~njas/sequences/A113401
>
> A113401 Algebraic degree of Cos[2Pi/n] for constructible n-gons 
> (A003401).   
> 1, 1, 1, 1, 2, 1, 2, 2, 2, 4, 4, 8, 4, 4, 4, 8, 8, 8, 8, 16, 8, 16, 
> 16, 16, 32,
> 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 64, 64, 128, 64, 64, 64, 64, 
> 64, 64, 64,
> 128, 128, 128, 128, 128, 128, 256, 128, 128, 128, 256, 256, 256, 256, 
> 512, 256,
> 256, 256, 256, 256, 256, 512
>
> OFFSET 1,5
> KEYWORD nonn,easy,nice
> AUTHOR E. W. Weisstein (eric(AT)weisstein.com), Oct 28, 2005
>
> I just wonder, isn't this possible at all to compute/estimate for 
> non-constructible
> polygons (heptagon, 9-gon, etc.)?
> (Some holes in my higher algebra, I guess.) From Weisstein's web page:
> "The angles m*Pi/n (with m,n integers) for which the trigonometric 
> functions
> may be expressed in terms of finite root extraction of real numbers are
> limited to values of m which are precisely those which produce 
> constructible polygons."
>
Thanks to Russ Cox's latest addition, I can now enter a query like this:
1,1,1,1,2,1,_,2,_,2,_,2,_,_,4,4,8,_,_,4,_,_,_,4
(where _'s occur at the positions of non-constructible polygons, 7, 9, 
11, 13, 14, etc.)
and other terms are from A113401. What I get?

http://www.research.att.com/~njas/sequences/A010554
NAME: phi(phi(n)), where phi is Euler totient function.
1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 4, 8, 2, 6, 4, 4, 4,
10, 4, 8, 4, 6, 4, 12, 4, 8, 8, 8, 8, 8, 4, 12, 6, 8, 8, 16, 4, 12,
8, 8, 10, 22, 8, 12, 8, 16, 8, 24, 6, 16, 8, 12, 12, 28, 8, 16, 8,
12, 16, 16, 8, 20, 16, 20, 8, 24, 8

(i.e. the number of primitive roots of n, when n has any.)
Hmm...

Not bothering you any more with this, before somebody says something wiser.

Yours,

Same.