A197366 - OEIS

%I #19 Oct 23 2024 00:53:52

%S 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,

%T 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,

%U 1,1,0,6,0,1,1,0,0,2,0,5,1,1,1,2,0,1

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

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

%o (GAP)

%o B:=[]; LoadPackage("sonata");

%o for m in [1..86] do

%o     n := 2*m; S:=[];

%o     for i in DivisorsInt(n)+1 do

%o         if IsPrime(i)=true then

%o             S:=Concatenation(S,[i]);

%o         fi;

%o     od;

%o     T:=[];

%o     for k in [1..Size(S)] do

%o         T:=Concatenation(T,[S[k]/(S[k]-1)]);

%o     od;

%o     max := n*Product(T); R:=[];

%o     for r in [1..Int(max)] do

%o         if Phi(r)=n then

%o             R:=Concatenation(R,[r]);

%o         fi;

%o     od;

%o     A:=[];

%o     for t in [1..NrSmallGroups(n)] do

%o         if IsAbelian(SmallGroup(n,t))=true then

%o             A:=Concatenation(A,[SmallGroup(n,t)]);

%o         fi;

%o     od;

%o     U:=[];

%o     for s in [1..Size(R)] do

%o         U:=Concatenation(U,[Units(Integers mod R[s])]);

%o     od;

%o     V:=[];

%o     for v in [1..Size(A)] do

%o         for w in [1..Size(U)] do

%o             if IsIsomorphicGroup(A[v],U[w])=true then

%o                 V:=Concatenation(V,[v]);

%o                 break;

%o             fi;

%o         od;

%o     od;

%o     B:=Concatenation(B,[Size(V)]);

%o od;

%o Print(B); # _Miles Englezou_, Oct 22 2024

%Y Cf. A101872, A179229.

%K nonn

%O 1,2

%A _Artur Jasinski_, Oct 14 2011

%E Name corrected by _Andrey Zabolotskiy_, Oct 21 2024

%E Terms a(17) onwards from _Miles Englezou_, Oct 22 2024