This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A093819 #40 Feb 16 2025 08:32:53
%S A093819 1,1,2,1,4,2,6,2,6,4,10,1,12,6,8,4,16,6,18,2,12,10,22,4,20,12,18,3,28,
%T A093819 8,30,8,20,16,24,3,36,18,24,8,40,12,42,5,24,22,46,8,42,20,32,6,52,18,
%U A093819 40,12,36,28,58,4,60,30,36,16,48,20,66,8,44,24,70,12,72,36,40,9,60,24
%N A093819 Algebraic degree of sin(2*Pi/n).
%C A093819 The degree formula given in the I. Niven reference on p. 37-8 (see below) appears as part of theorem 3.9 attributed to D. H. Lehmer. However, this part, concerning sin(2*Pi/n), differs from Lehmer's result, which in fact is incorrect. - _Wolfdieter Lang_, Jan 09 2011
%C A093819 This is also the algebraic degree of the area of a regular n-gon inscribed in the unit circle. - _Jack W Grahl_, Jan 10 2011
%C A093819 Every degree appears in this sequence except for the half-nontotients, A079695. - _T. D. Noe_, Jan 12 2011
%C A093819 See A181872/A181873 for the monic rational minimal polynomial of sin(2*Pi/n), and A181871 for the non-monic integer version. In A231188 the (monic and integer) minimal polynomials for 2*sin(2*Pi/n) are given. - _Wolfdieter Lang_, Nov 30 2013
%D A093819 I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.
%H A093819 T. D. Noe, <a href="/A093819/b093819.txt">Table of n, a(n) for n = 1..1000</a>
%H A093819 Sameen Ahmed Khan, <a href="https://doi.org/10.13189/ms.2021.090605">Trigonometric Ratios Using Algebraic Methods</a>, Mathematics and Statistics (2021) Vol. 9, No. 6, 899-907.
%H A093819 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrigonometryAngles.html">Trigonometry Angles</a>
%F A093819 a(4)=1, a(n)=phi(n) if gcd(n,8)<4; a(n)=phi(n)/4 if gcd(n,8)=4, and a(n)=phi(n)/2 if gcd(n,8)>4. Here phi(n)=A000010(n) (Euler totient). See the I. Niven reference, Theorem 3.9, p. 37-8. - _Wolfdieter Lang_, Jan 09 2011
%F A093819 a(n) = delta(c(n)/2) if c(n) = A178182(n) is even, and delta(c(n)) if c(n) is odd, with delta(n) = A055034(n), the degree of the algebraic number 2*cos(Pi/n). - _Wolfdieter Lang_, Nov 30 2013
%t A093819 a[4]=1; a[n_] := Module[{g=GCD[n, 8], e=EulerPhi[n]}, If[g<4, e, If[g==4, e/4, e/2]]]; Array[a, 1000]
%t A093819 f[n_] := Exponent[ MinimalPolynomial[ Sin[ 2Pi/n]][x], x]; Array[f, 75] (* _Robert G. Wilson v_, Jul 28 2014 *)
%Y A093819 Cf. A055035, A023022 (alg. degree of cos(2*Pi/n)), A183919.
%Y A093819 Cf. A181872/A181873, A181871, A231188. - _Wolfdieter Lang_, Nov 30 2013
%K A093819 nonn
%O A093819 1,3
%A A093819 _Eric W. Weisstein_, Apr 16 2004