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

%I A062771 #30 Dec 30 2022 03:55:01
%S A062771 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,
%T A062771 72,18,48,28,48,30,64,20,96,24,48,36,108,24,64,40,72,42,80,24,132,46,
%U A062771 64,42,120,32,96,52,108,40,96,36,168,58,64,60,180,36,128,48,120,66,128,44
%N A062771 Order of automorphism group of the group C_n X C_2 (where C_n is the cyclic group with n elements).
%H A062771 David A. Corneth, <a href="/A062771/b062771.txt">Table of n, a(n) for n = 1..10000</a>
%F A062771 For odd n: a(n) = phi(n) (sequence A000010).
%F A062771 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
%F A062771 Conjecture confirmed. - _Christian G. Bower_, May 20 2005
%F A062771 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
%F A062771 Sum_{k=1..n} a(k) ~ c * n^2, where c = 7/Pi^2 = 0.709248... . - _Amiram Eldar_, Oct 30 2022
%F A062771 Dirichlet g.f.: (zeta(s-1)/zeta(s))*((2^s-4/2^s+4)/(2^s-1)). - _Amiram Eldar_, Dec 30 2022
%t A062771 a[n_] := Switch[Mod[n, 4], 0, 4, 2, 6, _, 1]*EulerPhi[n];
%t A062771 Array[a, 69] (* _Jean-François Alcover_, Apr 19 2018 *)
%o A062771 (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
%Y A062771 Cf. A000010.
%K A062771 nonn,easy,mult
%O A062771 1,2
%A A062771 Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 18 2001
%E A062771 More terms from _Christian G. Bower_, May 20 2005