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

1,2

Links

Table of n, a(n) for n=1..86.

Formula

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

Prog

(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

Crossrefs

Cf. A101872, A179229.

Sequence in context: A350223 A274820 A230583 * A245715 A337736 A047885

Adjacent sequences: A197363 A197364 A197365 * A197367 A197368 A197369

Keyword

nonn

Author

Artur Jasinski, Oct 14 2011

Extensions

Name corrected by Andrey Zabolotskiy, Oct 21 2024

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

Status

approved