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

1,3

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.

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)

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).

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A326068 A318878 A306408 * A375124 A322250 A175542

Adjacent sequences: A232623 A232624 A232625 * A232627 A232628 A232629

Keyword

nonn,easy

Author

Wolfdieter Lang, Dec 12 2013

Status

approved