A060710 Number of subgroups of dihedral group with 2n elements, counting conjugate subgroups only once, i.e., conjugacy classes of subgroups of the dihedral group.

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, 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, 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

Offset: 1

Author

Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001

Keywords

• nonn

Comments

The total number of subgroups, counting conjugate subgroups as distinct, is A007503.

Also the number of subgroups of the group C_n x C_2 (where C_n is the cyclic group with n elements).

Examples

			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.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001622, A007503, A062011.

A row of A216624.

Programs

Mathematica

a[n_] := DivisorSum[n, 3-Mod[#,2]&];
Array[a, 100] (* Jean-François Alcover, Jun 03 2019 *)

PARI

a(n)=if(n<1, 0, sumdiv(n,d, 3-d%2)) /* Michael Somos, Sep 20 2005 */

PARI

{ for (n=1, 1000, write("b060710.txt", n, " ", sumdiv(n, d, 3 - d%2)); ) } \\ Harry J. Smith, Jul 10 2009

Sage

def A060710(n): return add(3 - int(is_odd(d)) for d in divisors(n))
[A060710(n) for n in (1..83)] # Peter Luschny, Sep 12 2012

Formula

For even n, a(n) = 2*tau(n) + tau(n/2).

For odd n, a(n) = tau(2n) = 2*tau(n) = 2*A000005(n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 12 2001

From Michael Somos, Sep 20 2005: (Start)

Moebius transform is period 2 sequence [2, 3, ...].

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)

a(n) = 4*tau(n) - tau(2n). - Ridouane Oudra, Jan 16 2023

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

Extensions

More terms from Vladeta Jovovic, Jul 15 2001