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

1,2

Comments

Maximal subgroups are counted.

Smallest such subgroup is the Frattini subgroup.

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

Links

Table of n, a(n) for n=1..70.

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

Formula

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

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 *)

Prog

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

Crossrefs

Cf. A007503.

Sequence in context: A359179 A278713 A200623 * A378211 A343442 A232635

Adjacent sequences: A248668 A248669 A248670 * A248672 A248673 A248674

Keyword

nonn

Author

Nandor Sieben, Oct 11 2014

Extensions

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

Status

approved