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

1,1

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).

Links

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

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A001622, A007503, A062011.

A row of A216624.

Sequence in context: A111570 A057954 A155896 * A271853 A146101 A206256

Adjacent sequences: A060707 A060708 A060709 * A060711 A060712 A060713

Keyword

nonn

Author

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

Extensions

More terms from Vladeta Jovovic, Jul 15 2001

Status

approved