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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 (list; graph; refs; listen; history; text; internal format)
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)my-deja.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)/(1-x^(2k)) = Sum_{k>0} 2*x^(2k-1)/(1-x^(2k-1))+3*x^(2k)/(1-x^(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 3-cycles. So a(3) = 4.

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. 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

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Last modified May 22 18:53 EDT 2019. Contains 323481 sequences. (Running on oeis4.)