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A179229
a(n) = number of Abelian groups of order 2*n which are not isomorphic to the group of units of the ring Z/kZ for any k.
1
0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 1, 0, 3, 1, 1, 1, 0, 0, 1, 0, 2, 2, 1, 2, 2, 0, 0, 1, 6, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 1, 0, 2, 1, 1, 2, 2, 3, 0, 2, 0, 3, 0, 2, 1, 1, 1, 1, 1, 2, 1, 10, 0, 1, 1, 2, 0, 1, 1, 4, 1, 1, 1, 3, 1, 0, 1, 2, 4, 1, 0, 1, 1, 1
OFFSET
1,8
COMMENTS
The group of units U of the ring Z/kZ is always an abelian group. - Miles Englezou, Oct 22 2024
FORMULA
a(n) + A197366(n) = A000688(2n). - R. J. Mathar, Jun 14 2019
EXAMPLE
a(1) = 0 because there is 1 Abelian group of order 2*1 = 2, and 3 distinct k such that the group of units U of Z/kZ has order 2; since U is always abelian, therefore every such U is isomorphic to C2.
a(4) = 1 because there are 3 Abelian groups of order 2*4 = 8, and 5 distinct k such that the group of units U of Z/kZ has order 8. The cyclic C8 is not isomorphic to any of these U.
a(8) = 2 because there are 5 Abelian groups of order 2*8 = 16, and 6 distinct k such that the group of units U of Z/kZ has order 16. The group C8 x C2 and the elementary abelian C2 x C2 x C2 x C2 are not isomorphic to any of these U.
PROG
(GAP)
B:=[]; LoadPackage("sonata");
for m in [1..64] do
n := 2*m;
S:=[];;
for i in DivisorsInt(n)+1 do
if IsPrime(i)=true then
S:=Concatenation(S, [i]);
fi;
od;
T:=[];
for k in [1..Size(S)] do
T:=Concatenation(T, [S[k]/(S[k]-1)]);
od;
max := n*Product(T);
R:=[];
for r in [1..Int(max)] do
if Phi(r)=n then
R:=Concatenation(R, [r]);
fi;
od;
A:=[];
for t in [1..NrSmallGroups(n)] do
if IsAbelian(SmallGroup(n, t))=true then
A:=Concatenation(A, [SmallGroup(n, t)]);
fi;
od;
U:=[];
for s in [1..Size(R)] do
U:=Concatenation(U, [Units(Integers mod R[s])]);
od;
V:=[];
for v in [1..Size(A)] do
for w in [1..Size(U)] do
if IsIsomorphicGroup(A[v], U[w])=true then
V:=Concatenation(V, [v]);
break;
fi;
od;
od;
B:=Concatenation(B, [Size(A)-Size(V)]);
od;
Print(B); # Miles Englezou, Oct 22 2024
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Artur Jasinski, Jul 03 2010
EXTENSIONS
Edited by Andrey Zabolotskiy and Miles Englezou, Oct 22 2024
Terms a(17) onwards from Miles Englezou, Oct 22 2024
STATUS
approved