OFFSET
1,2
COMMENTS
a(n) is also the number of non-congruent solutions to x^2 - y^2 == 1 (mod n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 21 2003
a(n) is the size of a square companion matrix of the minimal cyclotomic polynomial of (-1)^(1/n). - Eric Desbiaux, Dec 08 2015
a(n) is the degree of the (2n)-th cyclotomic field Q(zeta_(2n)). Note that Q(zeta_n) = Q(zeta_(2n)) for odd n. - Jianing Song, May 17 2021
REFERENCES
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, p. 28.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from Vincenzo Librandi)
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).
László Tóth, Counting solutions of quadratic congruences in several variables revisited, arXiv preprint arXiv:1404.4214 [math.NT], 2014.
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.
Wikipedia, Ramanujan's sum.
FORMULA
a(n) = Sum_{d|n and d is odd} n/d*mu(d).
Multiplicative with a(2^e) = 2^e and a(p^e) = p^e-p^(e-1), p>2.
Dirichlet g.f.: zeta(s-1)/zeta(s)*2^s/(2^s-1). - Ralf Stephan, Jun 17 2007
a(n) = A000010(2*n).
a(n) = phi(n)*(1+((n+1) mod 2)). - Gary Detlefs, Jul 13 2011
a(n) = A173557(n)*b(n) where b(n) = 1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 4, 1, 2, ... is the multiplicative function defined by b(p^e) = p^(e-1) if p<>2 and b(2^e)=2^e. b(n) = n/A204455(n). - R. J. Mathar, Jul 02 2013
a(n) = -c_{2n}(n) where c_q(n) is Ramanujan's sum. - Michael Somos, Aug 23 2013
a(n) = A055034(2*n), for n >= 2. - Wolfdieter Lang, Nov 30 2013
O.g.f.: Sum_{n >= 1} mu(2*n-1)*x^(2*n-1)/(1 - x^(2*n-1))^2. - Peter Bala, Mar 17 2019
a(n) = A000010(4*n)/2, for n > = 1 (see Apostol, Theorem 2.5, (b), p. 28). - Wolfdieter Lang, Nov 17 2019
a(n) = n - Sum_{d|n, n/d odd, d < n} a(d). - Ilya Gutkovskiy, May 30 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} A209229(gcd(n,k)).
a(n) = Sum_{k=1..n} A209229(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = 4/Pi^2 = 0.405284... (A185199). - Amiram Eldar, Oct 22 2022
MAPLE
[phi(2*n)$n=1..80]; # Muniru A Asiru, Mar 18 2019
MATHEMATICA
Table[EulerPhi[2 n], {n, 80}] (* Vincenzo Librandi, Aug 23 2013 *)
PROG
(PARI) a(n) = eulerphi(2*n)
(Sage) [euler_phi(2*n) for n in range(1, 74)] # Zerinvary Lajos, Jun 06 2009
(Python)
from sympy import totient
def A062570(n): return totient(n) if n&1 else totient(n)<<1 # Chai Wah Wu, Aug 04 2024
CROSSREFS
Row 1 of A379010.
KEYWORD
mult,nonn,easy
AUTHOR
Jason Earls, Jul 03 2001
EXTENSIONS
Corrected by Vladeta Jovovic, Dec 04 2002
STATUS
approved