  In this paper, we investigate two variations on the so-called persistence
problem of Sloane: the shifted version, which was introduced by Wagstaff; and
the nonzero version, proposed by Erd\H{o}s. We explore connections between
these problems and a recent conjecture of de Faria and Tresser regarding
equidistribution of the digits of some integer sequences and some of its
natural generalizations.
