A062771 Order of automorphism group of the group C_n X C_2 (where C_n is the cyclic group with n elements).

1, 6, 2, 8, 4, 12, 6, 16, 6, 24, 10, 16, 12, 36, 8, 32, 16, 36, 18, 32, 12, 60, 22, 32, 20, 72, 18, 48, 28, 48, 30, 64, 20, 96, 24, 48, 36, 108, 24, 64, 40, 72, 42, 80, 24, 132, 46, 64, 42, 120, 32, 96, 52, 108, 40, 96, 36, 168, 58, 64, 60, 180, 36, 128, 48, 120, 66, 128, 44

Offset: 1

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 18 2001

Keywords

• nonn
• easy
• mult

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010.

Programs

Mathematica

a[n_] := Switch[Mod[n, 4], 0, 4, 2, 6, _, 1]*EulerPhi[n];
Array[a, 69] (* Jean-François Alcover, Apr 19 2018 *)

PARI

a(n)=my(p=eulerphi(n)); if(n%2==1, p, if(n%4==2,6*p,4*p)); \\ Joerg Arndt, Sep 09 2020

Formula

For odd n: a(n) = phi(n) (sequence A000010).

Conjecture: a(n) = 6*phi(n) if n mod 4 = 2 and a(n) = 4*phi(n) if n mod 4 = 0. - Vladeta Jovovic, Jul 20 2001

Conjecture confirmed. - Christian G. Bower, May 20 2005

Multiplicative with a(2) = 6, a(2^e) = 2^(e+1), e>1, a(p^e) = (p-1)*p^(e-1), p>2. - Christian G. Bower, May 18 2005

Sum_{k=1..n} a(k) ~ c * n^2, where c = 7/Pi^2 = 0.709248... . - Amiram Eldar, Oct 30 2022

Dirichlet g.f.: (zeta(s-1)/zeta(s))*((2^s-4/2^s+4)/(2^s-1)). - Amiram Eldar, Dec 30 2022

Extensions

More terms from Christian G. Bower, May 20 2005