A062570 a(n) = phi(2*n).

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

Offset: 1

Author

Jason Earls, Jul 03 2001

Keywords

• mult
• nonn
• easy

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

The number of integers k from 1 to n such that gcd(n,k) is a power of 2. - Amiram Eldar, May 18 2025

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.

Crossrefs

Cf. A000010, A000034, A008683, A037225, A060968, A062803, A185199, A209229.

Column 1 of A129559, column 2 of A372673.

Row 1 of A379010.

Row sums of A129558 and of A129564.

Programs

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 *)

PARI

a(n) = eulerphi(2*n)

Python

from sympy import totient
def A062570(n): return totient(n) if n&1 else totient(n)<<1 # Chai Wah Wu, Aug 04 2024

Sage

[euler_phi(2*n) for n in range(1,74)] # Zerinvary Lajos, Jun 06 2009

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

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

a(n) = A000034(n) * A000010(n). - Amiram Eldar, May 18 2025

Extensions

Corrected by Vladeta Jovovic, Dec 04 2002