cp's OEIS Frontend

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

%I A248671 #21 Aug 29 2018 16:56:05
%S A248671 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 A248671 5,27,31,79,33,4,51,57,51,15,39,63,59,21,43,103,45,39,27,75,49,15,9,
%U A248671 21,75,45,55,15,75,27,83,93,61,79,63,99,35,4,87,151,69,57,99,151
%N A248671 Number of subgroups of the dihedral group Dn that are intersections of some maximal subgroups.
%C A248671 Maximal subgroups are counted.
%C A248671 Smallest such subgroup is the Frattini subgroup.
%C A248671 These subgroups are called intersection subgroups in Ernst and Sieben link.
%H A248671 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 A248671 a(n) = A007503(n) - 1 for squarefree n. - _Andrew Howroyd_, Jul 02 2018
%t A248671 a[n_] := With[{f = FactorInteger[n][[All, 1]]}, Sum[d+1, {d, Divisors[Times @@ f]}]-1];
%t A248671 Array[a, 70] (* _Jean-François Alcover_, Aug 29 2018, after _Andrew Howroyd_ *)
%o A248671 (GAP)
%o A248671 for n in [1..22] do
%o A248671   G:=DihedralGroup(2*n);
%o A248671   Ge:=Elements(G);
%o A248671   mse:=List(MaximalSubgroups(G),s->List(s,el->Position(Ge,el)));
%o A248671   C:=Combinations(mse);
%o A248671   Remove(C,1); # empty intersection is removed
%o A248671   I:=List(C,Intersection);
%o A248671   Sort(I);
%o A248671   I:=Unique(I);
%o A248671   Print(Size(I),",");
%o A248671 od;
%o A248671 (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 A248671 Cf. A007503.
%K A248671 nonn
%O A248671 1,2
%A A248671 _Nandor Sieben_, Oct 11 2014
%E A248671 a(23)-a(70) from _Andrew Howroyd_, Jul 02 2018