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

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, 5, 27, 31, 79, 33, 4, 51, 57, 51, 15, 39, 63, 59, 21, 43, 103, 45, 39, 27, 75, 49, 15, 9, 21, 75, 45, 55, 15, 75, 27, 83, 93, 61, 79, 63, 99, 35, 4, 87, 151, 69, 57, 99, 151

Offset: 1

Author

Nandor Sieben, Oct 11 2014

Keywords

• nonn

Comments

Maximal subgroups are counted.

Smallest such subgroup is the Frattini subgroup.

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

Links

• Dana C. Ernst, Nandor Sieben, Impartial achievement and avoidance games for generating finite groups, arXiv:1407.0784 [math.CO], 2014.

Crossrefs

Cf. A007503.

Programs

GAP

for n in [1..22] do
  G:=DihedralGroup(2*n);
  Ge:=Elements(G);
  mse:=List(MaximalSubgroups(G),s->List(s,el->Position(Ge,el)));
  C:=Combinations(mse);
  Remove(C,1); # empty intersection is removed
  I:=List(C,Intersection);
  Sort(I);
  I:=Unique(I);
  Print(Size(I),",");
od;

Mathematica

a[n_] := With[{f = FactorInteger[n][[All, 1]]}, Sum[d+1, {d, Divisors[Times @@ f]}]-1];
Array[a, 70] (* Jean-François Alcover, Aug 29 2018, after Andrew Howroyd *)

PARI

a(n) = my(f=factor(n)[,1]); sumdiv(prod(i=1, #f, f[i]), d, d+1 ) - 1; \\ Andrew Howroyd, Jul 02 2018

Formula

a(n) = A007503(n) - 1 for squarefree n. - Andrew Howroyd, Jul 02 2018

Extensions

a(23)-a(70) from Andrew Howroyd, Jul 02 2018