A113402 Algebraic degree of cos(Pi/n) for constructible n-gons (A003401).

1, 1, 1, 2, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 128, 128, 128, 128, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 512, 512, 512, 512, 512, 512, 512, 512, 512

Offset

1,4

Comments

a(n) is always a power of 2.

It would appear that a(n) <= a(n+1) and that for a(n)=2^k, the count for k beginning with 0 is 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, ...; or that the count for k is k+2 for k > 0. - Robert G. Wilson v, Jul 31 2014

Apparently v_2(a(n)) = A052146(n-1) for n >= 2 where v_2 is the 2-adic valuation. - Joerg Arndt, Jul 29 2014 [incorrect for n >= 561, Joerg Arndt, Mar 03 2019]

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..1632

Eric Weisstein's World of Mathematics, Trigonometry Angles

Mathematica

f[n_] := Exponent[MinimalPolynomial[Cos[Pi/n]][x], x]; Table[ f@ n, {n, Select[ Range@ 1300, IntegerQ[ Log[2, EulerPhi[#]]] &]}] (* Robert G. Wilson v, Jul 28 2014 *)
A092506 = {2, 3, 5, 17, 257, 65537}; s = Sort[Times @@@ Subsets@ A092506]; mx = 2500; t = Union@ Flatten@ Table[(2^n)*s[[i]], {i, 64}, {n, 0, Log2[mx/s[[i]]]}]; f[n_] := EulerPhi[ 2n]/2; f[1] = 1; f@# & /@ t (* Robert G. Wilson v, Jul 28 2014 *)

Crossrefs

Cf. A000217, A002024, A003401, A055034, A113401.

Sequence in context: A287654 A355911 A369207 * A238726 A240321 A054861

Adjacent sequences: A113399 A113400 A113401 * A113403 A113404 A113405

Keyword

nonn,easy,nice

Author

Eric W. Weisstein, Oct 28 2005

Status

approved