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 A232626 #27 May 16 2025 02:11:47
%S A232626 1,1,2,1,4,2,6,1,6,4,10,2,12,6,8,2,16,6,18,4,12,10,22,1,20,12,18,6,28,
%T A232626 8,30,4,20,16,24,6,36,18,24,2,40,12,42,10,24,22,46,4,42,20,32,12,52,
%U A232626 18,40,3,36,28,58,8,60,30,36,8,48,20,66,16,44,24,70,3,72,36,40
%N A232626 Degree of the algebraic number 2*sin(4*Pi/n).
%C A232626 See the comment on A231190 for the formula for 2*sin(Pi*4/n) = 2*cos(Pi*p(2,n)/q(2,n)) with gcd(p(2,n),q(2,n)) = 1, where p(2,n) = A231190(n) and q(2,n) = A232625(n). This shows that 2*sin(Pi*4/n) is an integer in the algebraic number field Q(rho(q(2,n))) of degree a(n) = delta(q(2,n)) with delta(k) = A055034(k).
%C A232626 This degree a(n) is given by I. Niven's Theorem 3.9, pp. 37-38, by Niven(n/gcd(2,n)) with Niven(n) = A093819(n) the degree of 2*sin(2*Pi/n). Note that Niven uses gcd(k, n) = 1 in the derivation, and Niven(4) = 1. See the bisection given in the formula section which is obtained from this.
%D A232626 Ivan Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.
%H A232626 Amiram Eldar, <a href="/A232626/b232626.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..175 from Vincenzo Librandi)
%F A232626 a(n) = delta(A232625(n)), n >=1, with delta(1) = 1 and delta(k) = phi(2*k)/2 with Euler's totient function phi (A000010). delta(k) = A055034(k).
%F A232626 a(2*k+1) = A093819(2*k+1), k >= 0.
%F A232626 For k >= 1: a(2*k) = A093819(k), that is a(2*k) = 1 if k=4, phi(k) if k odd or k == 2 (mod 4), phi(k)/2 if k == 0 (mod 8), phi(k)/4 if k == 4 (mod 8) (but not k=4).
%e A232626 a(1) = A093819(1) = 1; a(4) = phi(2) = 1; a(6) = phi(3) = 2; a(8) = 1; a(9) = A093819(9) = 6.
%t A232626 f[n_] := Exponent[ MinimalPolynomial[ 2Sin[ 4Pi/n]][x], x]; Array[f, 75] (* _Robert G. Wilson v_, Jul 28 2014 *)
%o A232626 (PARI) a(n) = {my(k = denominator((n-8)/(2*n))); if(k == 1, 1, eulerphi(2*k)/2);} \\ _Amiram Eldar_, Nov 09 2024
%Y A232626 Cf. A000010, A055034, A231190, A232625, A093819.
%K A232626 nonn,easy
%O A232626 1,3
%A A232626 _Wolfdieter Lang_, Dec 12 2013