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%I #45 Jan 21 2023 03:19:43
%S 2,5,4,8,4,10,4,11,6,10,4,16,4,10,8,14,4,15,4,16,8,10,4,22,6,10,8,16,
%T 4,20,4,17,8,10,8,24,4,10,8,22,4,20,4,16,12,10,4,28,6,15,8,16,4,20,8,
%U 22,8,10,4,32,4,10,12,20,8,20,4,16,8,20,4,33,4,10,12,16,8,20,4,28,10,10,4
%N Number of subgroups of dihedral group with 2n elements, counting conjugate subgroups only once, i.e., conjugacy classes of subgroups of the dihedral group.
%C The total number of subgroups, counting conjugate subgroups as distinct, is A007503.
%C Also the number of subgroups of the group C_n x C_2 (where C_n is the cyclic group with n elements).
%H Harry J. Smith, <a href="/A060710/b060710.txt">Table of n, a(n) for n = 1..1000</a>
%F For even n, a(n) = 2*tau(n) + tau(n/2).
%F For odd n, a(n) = tau(2n) = 2*tau(n) = 2*A000005(n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 12 2001
%F From _Michael Somos_, Sep 20 2005: (Start)
%F Moebius transform is period 2 sequence [2, 3, ...].
%F G.f.: Sum_{k>0} x^k*(2+3x^k)/(1-x^(2k)) = Sum_{k>0} 2*x^(2k-1)/(1-x^(2k-1)) + 3*x^(2k)/(1-x^(2k)). (End)
%F a(n) = 4*tau(n) - tau(2n). - _Ridouane Oudra_, Jan 16 2023
%F Sum_{k=1..n} a(k) ~ n*(5*log(n) + 10*gamma - log(2) - 5)/2, where gamma is Euler's constant (A001622). - _Amiram Eldar_, Jan 21 2023
%e The dihedral group D6 is isomorphic to the symmetric group S_3 and the conjugacy classes of subgroups are: the trivial group, the whole group, subgroup of order 2 generated by a transposition and the subgroup A_3 generated by the 3-cycles. So a(3) = 4.
%t a[n_] := DivisorSum[n, 3-Mod[#,2]&];
%t Array[a, 100] (* _Jean-François Alcover_, Jun 03 2019 *)
%o (PARI) a(n)=if(n<1, 0, sumdiv(n,d, 3-d%2)) /* _Michael Somos_, Sep 20 2005 */
%o (PARI) { for (n=1, 1000, write("b060710.txt", n, " ", sumdiv(n, d, 3 - d%2)); ) } \\ _Harry J. Smith_, Jul 10 2009
%o (Sage)
%o def A060710(n): return add(3 - int(is_odd(d)) for d in divisors(n))
%o [A060710(n) for n in (1..83)] # _Peter Luschny_, Sep 12 2012
%Y Cf. A001622, A007503, A062011.
%Y A row of A216624.
%K nonn
%O 1,1
%A Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
%E More terms from _Vladeta Jovovic_, Jul 15 2001
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