%I #18 Jul 19 2013 14:17:38

%S 1,2,24,4,120,24,336,8,72,120

%N Exponent of the group of Lipschitz quaternions in a reduced system modulo n.

%t cuaternios[n_] := Flatten[Table[{{a,-b,d ,-c},{b,a,-c,-d},{-d,c,a,-b},{c,d,b,a}},{a,n},{b,n},{c,n},{d,n}],3]; A227499[n_]:=Length@Select[cuaternios[n],GCD[Det[#],n]== 1 &]; cuater[n_] := Select[cuaternios[n], GCD[Det[#], n] == 1 &]; exp[1]=1; expo[M_,n_]:= Min@Select[Divisors@A227499[n],Mod[MatrixPower[M, #],n] == IdentityMatrix[4]&];a[n_] := lcm@Table[expo[cuater[n][[i]], n], {i, A227499[n]}]; lcm[lis_] := {aux = 1; Do[aux = LCM[aux, lis[[i]]], {i, 1, Length[lis]}]; aux}[[1]]; Table[a[n], {n,2,10}]

%Y Cf. A227499, A002322.

%K nonn,hard,more

%O 1,2

%A _José María Grau Ribas_, Jul 13 2013