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Number of subgroups of dihedral group with 2n elements, counting conjugate subgroups only once, i.e., conjugacy classes of subgroups of the dihedral group.
5

%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