%I
%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] (* _JeanFrançois 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
