cp's OEIS Frontend

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

%I A197366 #19 Oct 23 2024 00:53:52
%S A197366 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 A197366 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 A197366 1,1,0,6,0,1,1,0,0,2,0,5,1,1,1,2,0,1
%N A197366 Number of Abelian groups of order 2n which are isomorphic with the group of units of the ring Z/kZ for some k.
%F A197366 a(n) = A101872(n) - A179229(n).
%o A197366 (GAP)
%o A197366 B:=[]; LoadPackage("sonata");
%o A197366 for m in [1..86] do
%o A197366     n := 2*m; S:=[];
%o A197366     for i in DivisorsInt(n)+1 do
%o A197366         if IsPrime(i)=true then
%o A197366             S:=Concatenation(S,[i]);
%o A197366         fi;
%o A197366     od;
%o A197366     T:=[];
%o A197366     for k in [1..Size(S)] do
%o A197366         T:=Concatenation(T,[S[k]/(S[k]-1)]);
%o A197366     od;
%o A197366     max := n*Product(T); R:=[];
%o A197366     for r in [1..Int(max)] do
%o A197366         if Phi(r)=n then
%o A197366             R:=Concatenation(R,[r]);
%o A197366         fi;
%o A197366     od;
%o A197366     A:=[];
%o A197366     for t in [1..NrSmallGroups(n)] do
%o A197366         if IsAbelian(SmallGroup(n,t))=true then
%o A197366             A:=Concatenation(A,[SmallGroup(n,t)]);
%o A197366         fi;
%o A197366     od;
%o A197366     U:=[];
%o A197366     for s in [1..Size(R)] do
%o A197366         U:=Concatenation(U,[Units(Integers mod R[s])]);
%o A197366     od;
%o A197366     V:=[];
%o A197366     for v in [1..Size(A)] do
%o A197366         for w in [1..Size(U)] do
%o A197366             if IsIsomorphicGroup(A[v],U[w])=true then
%o A197366                 V:=Concatenation(V,[v]);
%o A197366                 break;
%o A197366             fi;
%o A197366         od;
%o A197366     od;
%o A197366     B:=Concatenation(B,[Size(V)]);
%o A197366 od;
%o A197366 Print(B); # _Miles Englezou_, Oct 22 2024
%Y A197366 Cf. A101872, A179229.
%K A197366 nonn
%O A197366 1,2
%A A197366 _Artur Jasinski_, Oct 14 2011
%E A197366 Name corrected by _Andrey Zabolotskiy_, Oct 21 2024
%E A197366 Terms a(17) onwards from _Miles Englezou_, Oct 22 2024