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A007503 Number of subgroups of dihedral group: sigma(n) + d(n).
(Formerly M1321)
37
2, 5, 6, 10, 8, 16, 10, 19, 16, 22, 14, 34, 16, 28, 28, 36, 20, 45, 22, 48, 36, 40, 26, 68, 34, 46, 44, 62, 32, 80, 34, 69, 52, 58, 52, 100, 40, 64, 60, 98, 44, 104, 46, 90, 84, 76, 50, 134, 60, 99, 76, 104, 56, 128, 76, 128, 84, 94, 62, 180, 64, 100, 110, 134 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Essentially first differences of A257644. - Franklin T. Adams-Watters, Nov 05 2015
Write D_{2n} as <a, x | a^n = x^2 = 1, x*a*x = a^(-1)>, then the subgroups are of the form <a^d> for d|n or <a^d, a^r*x> for d|n and 0 <= r < d. There are d(n) subgroups of the first type and sigma(n) subgroups of the second type. - Jianing Song, Jul 21 2022
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
David W. Jensen and Eric R. Bussian, A Number-Theoretic Approach to Counting Subgroups of Dihedral Groups, Two-Year College Math. Jnl., 23 (1992), 150-152.
The Group Properties Wiki, Subgroup structure of dihedral groups
FORMULA
G.f.: Sum_{k>=1} 1/(1-x^k)^2. - Benoit Cloitre, Apr 21 2003
G.f.: Sum_{i>=1} (1+i)*x^i/(1-x^i). - Jon Perry, Jul 03 2004
a(n) = Sum_{d|n} tau(p^d), where tau is A000005 and p any prime. - Enrique Pérez Herrero, Apr 14 2012
a(n) = Sum_{d divides n} d+1. - Joerg Arndt, Apr 14 2013
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(1+1/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018
a(n) = A000005(n) + A000203(n). - Omar E. Pol, Aug 19 2019
EXAMPLE
a(4) = 10 since D_8 = <a, x | a^4 = x^2 = 1, x*a*x = a^(-1)> has 10 subgroups. The 6 subgroups {e}, {e,a^2}, {e,a,a^2,a^3}, {e,a^2,x,a^2*x}, {e,a^2,a*x,a^3*x} and D_8 are normal, and the 4 subgroups {e,x}, {e,a*x}, {e,a^2*x} and {e,a^3*x} are not. - Jianing Song, Jul 21 2022
MAPLE
with(numtheory): seq(sigma(n)+tau(n), n=1..56) ; # Zerinvary Lajos, Jun 04 2008
MATHEMATICA
A007503[n_]:=DivisorSum[n, DivisorSigma[0, 2^#]&]; Array[A007503, 20] (* Enrique Pérez Herrero, Apr 14 2012 *)
PROG
(PARI) a(n) = sumdiv(n, d, d+1 ); \\ Joerg Arndt, Apr 14 2013
(Haskell)
a007503 = sum . map (+ 1) . a027750_row'
-- Reinhard Zumkeller, Nov 09 2015
CROSSREFS
Cf. A000005, A000203, A037852 (number of normal subgroups).
Cf. A027750, A257644 (cumulative sums, start=1).
Sequence in context: A226810 A054463 A295741 * A337298 A184418 A112967
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)