

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
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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, 899907.
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(2ni, 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



