A197366 Number of Abelian groups of order 2n which are isomorphic with the group of units of the ring Z/kZ for some k.

1, 2, 1, 2, 1, 2, 0, 3, 1, 2, 1, 2, 0, 1, 1, 4, 0, 3, 0, 3, 1, 1, 1, 3, 0, 1, 1, 1, 1, 2, 0, 5, 1, 0, 1, 5, 0, 0, 1, 3, 1, 1, 0, 3, 0, 1, 0, 5, 0, 1, 1, 1, 1, 3, 1, 3, 0, 1, 0, 2, 0, 0, 1, 5, 1, 1, 0, 1, 1, 1, 0, 6, 0, 1, 1, 0, 0, 2, 0, 5, 1, 1, 1, 2, 0, 1

Offset: 1

Author

Artur Jasinski, Oct 14 2011

Keywords

• nonn

Crossrefs

Cf. A101872, A179229.

Programs

GAP

B:=[]; LoadPackage("sonata");
for m in [1..86] 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(V)]);
od;
Print(B); # Miles Englezou, Oct 22 2024

Formula

a(n) = A101872(n) - A179229(n).

Extensions

Name corrected by Andrey Zabolotskiy, Oct 21 2024

Terms a(17) onwards from Miles Englezou, Oct 22 2024