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A248671 Number of subgroups of the dihedral group Dn that are intersections of some maximal subgroups. 0


%S 1,4,5,4,7,15,9,4,5,21,13,15,15,27,27,4,19,15,21,21,35,39,25,15,7,45,

%T 5,27,31,79,33,4,51,57,51,15,39,63,59,21,43,103,45,39,27,75,49,15,9,

%U 21,75,45,55,15,75,27,83,93,61,79,63,99,35,4,87,151,69,57,99,151

%N Number of subgroups of the dihedral group Dn that are intersections of some maximal subgroups.

%C Maximal subgroups are counted.

%C Smallest such subgroup is the Frattini subgroup.

%C These subgroups are called intersection subgroups in Ernst and Sieben link.

%H Dana C. Ernst, Nandor Sieben, <a href="http://arxiv.org/abs/1407.0784">Impartial achievement and avoidance games for generating finite groups</a>, arXiv:1407.0784 [math.CO], 2014.

%F a(n) = A007503(n) - 1 for squarefree n. - _Andrew Howroyd_, Jul 02 2018

%t a[n_] := With[{f = FactorInteger[n][[All, 1]]}, Sum[d+1, {d, Divisors[Times @@ f]}]-1];

%t Array[a, 70] (* _Jean-Fran├žois Alcover_, Aug 29 2018, after _Andrew Howroyd_ *)

%o (GAP)

%o for n in [1..22] do

%o G:=DihedralGroup(2*n);

%o Ge:=Elements(G);

%o mse:=List(MaximalSubgroups(G),s->List(s,el->Position(Ge,el)));

%o C:=Combinations(mse);

%o Remove(C,1); # empty intersection is removed

%o I:=List(C,Intersection);

%o Sort(I);

%o I:=Unique(I);

%o Print(Size(I),",");

%o od;

%o (PARI) a(n) = my(f=factor(n)[,1]); sumdiv(prod(i=1, #f, f[i]), d, d+1 ) - 1; \\ _Andrew Howroyd_, Jul 02 2018

%Y Cf. A007503.

%K nonn

%O 1,2

%A _Nandor Sieben_, Oct 11 2014

%E a(23)-a(70) from _Andrew Howroyd_, Jul 02 2018

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Last modified January 17 12:50 EST 2020. Contains 330958 sequences. (Running on oeis4.)