%N Start with i=1 and j=2. Concatenate i and j, get k = floor ij/j, concatenate j and k, etc.
%C It is easy to see that in all such sequences a(n+2) > a(n). Therefore for any pair of initial terms, the sequence consist of two interweaving strictly increasing subsequences formed by the elements at odd and even positions. So there are no loops in such sequences.
%o (PARI) a=[1,2]; for(k=3,50,a=concat(a,[eval(concat(Str(a[k-2]),Str(a[k-1])))\a[k-1]])); a
%A _Eric Angelini_, Aug 05 2007
%E More terms, PARI program and comment from Max Alekseyev, Aug 05 2007