A007503 Number of subgroups of dihedral group: sigma(n) + d(n).

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

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn

Comments

Essentially first differences of A257644. - Franklin T. Adams-Watters, Nov 05 2015

Write D_{2n} as , then the subgroups are of the form for d|n or 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

Inverse Möbius transform of n+1 (see Arndt formula). - Wesley Ivan Hurt, Jul 05 2025

Examples

			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
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Keith Conrad, Dihedral Groups II
• 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
• Index entries for sequences related to groups

Crossrefs

Cf. A000005, A000203, A037852 (number of normal subgroups).

Cf. A027750, A257644 (cumulative sums, start=1).

Programs

Haskell

a007503 = sum . map (+ 1) . a027750_row'
-- Reinhard Zumkeller, Nov 09 2015

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 *)

PARI

a(n) = sumdiv(n,d, d+1 ); \\ Joerg Arndt, Apr 14 2013

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

a(n) = A348219(n) + A386438(n). - Wesley Ivan Hurt, Jul 21 2025