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
KEYWORD
nonn
AUTHOR
Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
EXTENSIONS
More terms from Vladeta Jovovic, Jul 15 2001
STATUS
approved