cp's OEIS Frontend

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.

%I A179229 #42 Oct 25 2024 09:46:23
%S A179229 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,
%T A179229 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,
%U A179229 0,1,1,4,1,1,1,3,1,0,1,2,4,1,0,1,1,1
%N 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.
%C A179229 The group of units U of the ring Z/kZ is always an abelian group. - _Miles Englezou_, Oct 22 2024
%H A179229 Wikipedia, <a href="https://en.wikipedia.org/wiki/List_of_small_groups">List of small groups</a>.
%F A179229 a(n) + A197366(n) = A000688(2n). - _R. J. Mathar_, Jun 14 2019
%e A179229 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.
%e A179229 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.
%e A179229 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.
%o A179229 (GAP)
%o A179229 B:=[]; LoadPackage("sonata");
%o A179229 for m in [1..64] do
%o A179229     n := 2*m;
%o A179229     S:=[];;
%o A179229     for i in DivisorsInt(n)+1 do
%o A179229         if IsPrime(i)=true then
%o A179229             S:=Concatenation(S,[i]);
%o A179229         fi;
%o A179229     od;
%o A179229     T:=[];
%o A179229     for k in [1..Size(S)] do
%o A179229         T:=Concatenation(T,[S[k]/(S[k]-1)]);
%o A179229     od;
%o A179229     max := n*Product(T);
%o A179229     R:=[];
%o A179229     for r in [1..Int(max)] do
%o A179229         if Phi(r)=n then
%o A179229             R:=Concatenation(R,[r]);
%o A179229         fi;
%o A179229     od;
%o A179229     A:=[];
%o A179229     for t in [1..NrSmallGroups(n)] do
%o A179229         if IsAbelian(SmallGroup(n,t))=true then
%o A179229             A:=Concatenation(A,[SmallGroup(n,t)]);
%o A179229         fi;
%o A179229     od;
%o A179229     U:=[];
%o A179229     for s in [1..Size(R)] do
%o A179229         U:=Concatenation(U,[Units(Integers mod R[s])]);
%o A179229     od;
%o A179229     V:=[];
%o A179229     for v in [1..Size(A)] do
%o A179229         for w in [1..Size(U)] do
%o A179229             if IsIsomorphicGroup(A[v],U[w])=true then
%o A179229                 V:=Concatenation(V,[v]);
%o A179229                 break;
%o A179229             fi;
%o A179229         od;
%o A179229     od;
%o A179229 B:=Concatenation(B,[Size(A)-Size(V)]);
%o A179229 od;
%o A179229 Print(B); # _Miles Englezou_, Oct 22 2024
%Y A179229 Cf. A046660, A197366, A014197.
%K A179229 nonn
%O A179229 1,8
%A A179229 _Artur Jasinski_, Jul 03 2010
%E A179229 Edited by _Andrey Zabolotskiy_ and _Miles Englezou_, Oct 22 2024
%E A179229 Terms a(17) onwards from _Miles Englezou_, Oct 22 2024