The OEIS is supported by the many generous donors to the OEIS Foundation. Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A055034 a(1) = 1, a(n) = phi(2*n)/2 for n > 1. 88
 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 For n > 1, a(n) is the number of primitive Pythagorean triples (f,g,h) for which there exist positive integers n and k such that f = 2*n*k, g = n^2 - k^2, h = n^2 + k^2. Let U = {1,2,...,2*n-1}, V = {v element of U: v mod 2 = 0}, W = {w element of U\V: gcd(w,2*n) != 1} and X = {1,2,...,n-1}, Y = {y element of X: n == y (mod 2)}, Z = {z element of X\Y: gcd(z,n) != 1}. Then phi(2*n) = |U| - (|V| + |W|) = 2*n - 1 - (2*|Y| + 2*|Z| + 1) = 2*n - 2 - 2*|Y| - 2*|Z| and phi(2*n)/2 = n - 1 - |Y| - |Z|. This is equivalent to the number of primitive Pythagorean triples (f,g,h), where from n-1 pairs (n,k) the ones with n == k (mod 2) or gcd(n,k) != 1 have to be subtracted. - Felix Huber, Apr 17 2023 LINKS T. D. Noe, Table of n, a(n) for n = 1..2000 Sameen Ahmed Khan, Trigonometric Ratios Using Algebraic Methods, Mathematics and Statistics, Vol. 9, No. 6 (2021), 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. Wikipedia, Pythagorean triple, Euclid's formula. 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 G.f.: (x + Sum_{n>=1} mu(2n-1) * x^(2n-1) / (1-x^(2n-1))^2) / 2 . - Mamuka Jibladze, Dec 14 2022 Sum_{k=1..n} a(k) ~ c * n^2, where c = 2/Pi^2 = 0.202642... (A185197). - Amiram Eldar, Feb 11 2023 EXAMPLE a(10) = 4 since the primitive Pythagorean triples generated by Euclid's formula (n, k) -> [2*n*k, n^2 - k^2, n^2 + k^2] are: (10, 1) -> [20, 99, 101]; (10, 3) -> [60, 91, 109]; (10, 7) -> [140, 51, 149]; (10, 9) -> [180, 19, 181]. - Peter Luschny, Apr 16 2023 MAPLE with(numtheory); A055034:=n->ceil(phi(2*n)/2); seq(A055034(k), k=1..100); # Wesley Ivan Hurt, Oct 24 2013 a := n -> if n = 1 then 1 else iquo(NumberTheory:-Totient(2*n), 2) fi: seq(a(k), k = 1..100); # Peter Luschny, Apr 16 2023 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 (Python) from sympy import totient def A055034(n): return totient(n<<1)>>1 if n>1 else 1 # Chai Wah Wu, Nov 24 2023 CROSSREFS Cf. A000010, A094192, A185197, A187360, A222946. Sequence in context: A155940 A186963 A060473 * A362739 A112184 A112213 Adjacent sequences: A055031 A055032 A055033 * A055035 A055036 A055037 KEYWORD nonn,easy,changed 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 6 14:44 EST 2023. Contains 367609 sequences. (Running on oeis4.)