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.
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, 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, 0, 1, 1, 4, 1, 1, 1, 3, 1, 0, 1, 2, 4, 1, 0, 1, 1, 1
Offset: 1
Keywords
Examples
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. 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. 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.
Links
- Wikipedia, List of small groups.
Programs
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GAP
B:=[]; LoadPackage("sonata"); for m in [1..64] 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(A)-Size(V)]); od; Print(B); # Miles Englezou, Oct 22 2024
Formula
Extensions
Edited by Andrey Zabolotskiy and Miles Englezou, Oct 22 2024
Terms a(17) onwards from Miles Englezou, Oct 22 2024
Comments