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A055034 a(1) = 1, a(n) = phi(2*n)/2 for n>1. 84
1, 1, 1, 2, 2, 2, 3, 4, 3, 4, 5, 4, 6, 6, 4, 8, 8, 6, 9, 8, 6, 10, 11, 8, 10, 12, 9, 12, 14, 8, 15, 16, 10, 16, 12, 12, 18, 18, 12, 16, 20, 12, 21, 20, 12, 22, 23, 16, 21, 20, 16, 24, 26, 18, 20, 24, 18, 28, 29, 16, 30, 30, 18, 32, 24, 20, 33, 32, 22, 24, 35, 24, 36, 36, 20, 36, 30 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

For n>1, gives number of times n appears in A094192. - Lekraj Beedassy, Jun 04 2004

Number of positive integers less than n that are relatively prime to n, and have opposite parity to n, for n >= 2. a(1) = 1. - Anne M. Donovan (anned3005(AT)aol.com), Jul 18 2005 [rewritten by Wolfdieter Lang, Apr 08 2020]

Degree of minimal polynomial of cos(Pi/n) over the rationals. For the minimal polynomials of 2*cos(Pi/n), n>=1, see A187360. - Wolfdieter Lang, Jul 19 2011.

a(n) is, for n>=2, the number of (positive) odd numbers 2*k+1 < n satisfying gcd(2*k+1,n)=1. See the formula for the zeros of the minimal polynomials A187360. E.g., n=10: 1,3,7,9, hence a(10)=4. - Wolfdieter Lang, Aug 17 2011.

a(n) is, for n>=2, the number of nonzero entries in row n of the triangle A222946. See the Beedassy and Donovan comment . - Wolfdieter Lang, Mar 24 2013

Number of partitions of 2n into exactly two relatively prime parts. - Wesley Ivan Hurt, Dec 22 2013

For n > 1, a(n) is the number of pairs of complex embeddings of the (2n)-th cyclotomic field Q(zeta_(2n)) (there are no real embeddings). Note that Q(zeta_n) = Q(zeta_(2n)) for odd n. By Dirichlet's unit theorem, the group of units of Z[zeta_(2n)] is isomorphic to C_(2n) X Z^{a(n)-1}, where C_(2n) is the group of all (2n)-th roots of unity. - Jianing Song, May 17 2021

LINKS

T. D. Noe, Table of n, a(n) for n = 1..2000

Sameen Ahmed Khan, Trigonometric Ratios Using Algebraic Methods, Mathematics and Statistics (2021) Vol. 9, No. 6, 899-907.

Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.

Eric Weisstein's World of Mathematics, Trigonometry Angles

FORMULA

a(n) = ceiling( phi(2n)/2 ). - Wesley Ivan Hurt, Jun 16 2013

a(n) = Sum_{i=1..n} floor(1 / gcd(2n-i, i)). - Wesley Ivan Hurt, Dec 22 2013

MAPLE

with(numtheory); A055034:=n->ceil(phi(2*n)/2); seq(A055034(k), k=1..100); # Wesley Ivan Hurt, Oct 24 2013

MATHEMATICA

Join[{1}, EulerPhi[2*Range[2, 100]]/2] (* Harvey P. Dale, Aug 12 2011 *)

PROG

(PARI) a(n)=ceil(eulerphi(2*n)/2) \\ Charles R Greathouse IV, Feb 21 2013

CROSSREFS

Cf. A000010, A094192, A187360, A222946.

Sequence in context: A155940 A186963 A060473 * A112184 A112213 A238957

Adjacent sequences: A055031 A055032 A055033 * A055035 A055036 A055037

KEYWORD

nonn,easy

AUTHOR

Shawn Cokus (Cokus(AT)math.washington.edu)

EXTENSIONS

Better description from Benoit Cloitre, Feb 01 2002

Edited by Ray Chandler, Jul 20 2005

STATUS

approved

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Last modified December 3 09:39 EST 2022. Contains 358517 sequences. (Running on oeis4.)