OFFSET
1,2
COMMENTS
By Lagrange's Theorem a(n) is a multiple of n.
Are all terms after the first even?
The above conjecture is true. For even n, a(n) is even by Lagrange's theorem. For odd n, it follows from the fact that every dihedral group D_{2n} has a commutator subgroup of order n when n is odd; as no group of odd order is perfect, 2*n is the smallest possible order that such a commutator subgroup can be contained in. (For an extended proof see the Miles Englezou link.) - Miles Englezou, Mar 08 2024
LINKS
Miles Englezou, Table of n, a(n) for n = 1..255
Groupprops, Subgroup structure of dihedral groups.
Miles Englezou, Proof that A340518(n) is even.
FORMULA
a(2n+1) = 4n+2. - Miles Englezou, Mar 08 2024
EXAMPLE
The fourth term is 12, because 12 is the smallest order of a group G with |G'| = 4, A_4 being an example.
PROG
(GAP)
# Produces a list A of the first 255 terms
A:=[];
N:=[1..255];
F:=[1..20]; # for large n the array F may need to be extended beyond 20
for n in N do
for k in F do
L:=List([1..NrSmallGroups(n*k)], i->Size(DerivedSubgroup(SmallGroup(n*k, i))));;
if Positions(L, n)<>[] then
Add(A, n*k);
break;
fi;
od;
od; # Miles Englezou, Feb 26 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Des MacHale, Jan 24 2021
EXTENSIONS
More terms from Miles Englezou, Feb 26 2024
STATUS
approved