A232626 Degree of the algebraic number 2*sin(4*Pi/n).

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, 8, 30, 4, 20, 16, 24, 6, 36, 18, 24, 2, 40, 12, 42, 10, 24, 22, 46, 4, 42, 20, 32, 12, 52, 18, 40, 3, 36, 28, 58, 8, 60, 30, 36, 8, 48, 20, 66, 16, 44, 24, 70, 3, 72, 36, 40

Offset: 1

Author

Wolfdieter Lang, Dec 12 2013

Keywords

• nonn
• easy

Comments

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).

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.

Examples

			a(1) = A093819(1) = 1; a(4) = phi(2) = 1; a(6) = phi(3) = 2; a(8) = 1; a(9) = A093819(9) = 6.
		

References

• Ivan Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..175 from Vincenzo Librandi)

Crossrefs

Cf. A000010, A055034, A231190, A232625, A093819.

Programs

Mathematica

f[n_] := Exponent[ MinimalPolynomial[ 2Sin[ 4Pi/n]][x], x]; Array[f, 75] (* Robert G. Wilson v, Jul 28 2014 *)

PARI

a(n) = {my(k = denominator((n-8)/(2*n))); if(k == 1, 1, eulerphi(2*k)/2);} \\ Amiram Eldar, Nov 09 2024

Formula

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).

a(2*k+1) = A093819(2*k+1), k >= 0.

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).