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A248671 Number of subgroups of the dihedral group Dn that are intersections of some maximal subgroups. 0
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A021877 A278713 A200623 * A232635 A201296 A246954

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

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Last modified December 7 09:33 EST 2019. Contains 329843 sequences. (Running on oeis4.)