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
classes .
The **3x+1** function **T** is in the class .
H. Möller [54] 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 [41] proved the following quantitative version of this result,
thereby generalizing Theorem D.