Several authors have investigated the range of validity of the result that
has a finite stopping time for almost all integers n by considering more
general classes of periodicity linear functions.
One such class consists of all functions which
are given by
where m and d are positive integers with and ,
is a fixed set of residue class representatives of the nonzero residue
The 3x+1 function T is in the class .
H. Möller  completely characterized the
functions in the set which have a finite
stopping time for almost all integers n.
He showed they are exactly those functions for which
E. Heppner  proved the following quantitative version of this result,
thereby generalizing Theorem D.