

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
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OFFSET

1,4


COMMENTS

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*n1}, V = {v element of U: v mod 2 = 0}, W = {w element of U\V: gcd(w,2*n) != 1} and X = {1,2,...,n1}, 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 n1 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



FORMULA

G.f.: (x + Sum_{n>=1} mu(2n1) * x^(2n1) / (1x^(2n1))^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);
a := n > if n = 1 then 1 else iquo(NumberTheory:Totient(2*n), 2) fi:


MATHEMATICA

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


PROG

(Python)
from sympy import totient


CROSSREFS



KEYWORD

nonn,easy,changed


AUTHOR

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


EXTENSIONS



STATUS

approved



