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%I M1321 #62 Jul 21 2022 07:41:37
%S 2,5,6,10,8,16,10,19,16,22,14,34,16,28,28,36,20,45,22,48,36,40,26,68,
%T 34,46,44,62,32,80,34,69,52,58,52,100,40,64,60,98,44,104,46,90,84,76,
%U 50,134,60,99,76,104,56,128,76,128,84,94,62,180,64,100,110,134
%N Number of subgroups of dihedral group: sigma(n) + d(n).
%C Essentially first differences of A257644. - _Franklin T. Adams-Watters_, Nov 05 2015
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007503/b007503.txt">Table of n, a(n) for n = 1..1000</a>
%H David W. Jensen and Eric R. Bussian, <a href="http://www.jstor.org/stable/2686678">A Number-Theoretic Approach to Counting Subgroups of Dihedral Groups</a>, Two-Year College Math. Jnl., 23 (1992), 150-152.
%H The Group Properties Wiki, <a href="https://groupprops.subwiki.org/wiki/Subgroup_structure_of_dihedral_groups">Subgroup structure of dihedral groups</a>
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%F G.f.: Sum_{k>=1} 1/(1-x^k)^2. - _Benoit Cloitre_, Apr 21 2003
%F G.f.: Sum_{i>=1} (1+i)*x^i/(1-x^i). - _Jon Perry_, Jul 03 2004
%F a(n) = Sum_{d|n} tau(p^d), where tau is A000005 and p any prime. - _Enrique Pérez Herrero_, Apr 14 2012
%F a(n) = Sum_{d divides n} d+1. - _Joerg Arndt_, Apr 14 2013
%F 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
%F a(n) = A000005(n) + A000203(n). - _Omar E. Pol_, Aug 19 2019
%e 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
%p with(numtheory): seq(sigma(n)+tau(n), n=1..56) ; # _Zerinvary Lajos_, Jun 04 2008
%t A007503[n_]:=DivisorSum[n,DivisorSigma[0,2^#]&]; Array[A007503,20] (* _Enrique Pérez Herrero_, Apr 14 2012 *)
%o (PARI) a(n) = sumdiv(n,d, d+1 ); \\ _Joerg Arndt_, Apr 14 2013
%o (Haskell)
%o a007503 = sum . map (+ 1) . a027750_row'
%o -- _Reinhard Zumkeller_, Nov 09 2015
%Y Cf. A000005, A000203, A037852 (number of normal subgroups).
%Y Cf. A027750, A257644 (cumulative sums, start=1).
%K nonn
%O 1,1
%A _N. J. A. Sloane_, _Robert G. Wilson v_
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