

A062570


a(n) = phi(2*n).


19



1, 2, 2, 4, 4, 4, 6, 8, 6, 8, 10, 8, 12, 12, 8, 16, 16, 12, 18, 16, 12, 20, 22, 16, 20, 24, 18, 24, 28, 16, 30, 32, 20, 32, 24, 24, 36, 36, 24, 32, 40, 24, 42, 40, 24, 44, 46, 32, 42, 40, 32, 48, 52, 36, 40, 48, 36, 56, 58, 32, 60, 60, 36, 64, 48, 40, 66, 64, 44, 48, 70, 48, 72
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OFFSET

1,2


COMMENTS

a(n) is also the number of noncongruent 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, SpringerVerlag, 1976, p. 28.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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_{dn and d is odd} n/d*mu(d).
Multiplicative with a(2^e) = 2^e and a(p^e) = p^ep^(e1), p>2.
Dirichlet g.f.: zeta(s1)/zeta(s)*2^s/(2^s1).  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^(e1) 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*n1)*x^(2*n1)/(1  x^(2*n1))^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_{dn, n/d odd, d < n} a(d).  Ilya Gutkovskiy, May 30 2020
Dirichlet convolution of A000010 and A209229.  Werner Schulte, Jan 17 2021
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


CROSSREFS

Cf. A000010, A000034, A008683, A037225, A060968, A062803, A185199, A209229.
Sequence in context: A035114 A202103 A333787 * A108514 A317419 A120456
Adjacent sequences: A062567 A062568 A062569 * A062571 A062572 A062573


KEYWORD

mult,nonn,easy


AUTHOR

Jason Earls, Jul 03 2001


EXTENSIONS

Corrected by Vladeta Jovovic, Dec 04 2002


STATUS

approved



