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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
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
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 * A343442 A232635 A201296
KEYWORD
nonn
AUTHOR
Nandor Sieben, Oct 11 2014
EXTENSIONS
a(23)-a(70) from Andrew Howroyd, Jul 02 2018
STATUS
approved