

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.


5



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
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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 odd n: a(n) = tau(2n) = 2*tau(n)= 2*A000005(n).  Ahmed Fares (ahmedfares(AT)mydeja.com), Jul 12 2001
For even n, a(n) = 2*tau(n)+tau(n/2).
Moebius transform is period 2 sequence [2, 3, ...].  Michael Somos, Sep 20 2005
G.f.: Sum_{k>0} x^k(2+3x^k)/(1x^(2k)) = Sum_{k>0} 2*x^(2k1)/(1x^(2k1))+3*x^(2k)/(1x^(2k)).  Michael Somos, Sep 20 2005


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 3cycles. So a(3) = 4.


PROG

(PARI) a(n)=if(n<1, 0, sumdiv(n, d, 3d%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. 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



