Let
be a positive
integer and
be the sum of the squares of its decimal digits. When there exists a positive integer
such that
the
-th
iterate of
on
is 1,
i.e.,
,
then
is called a happy number. The notion of happy numbers has been
generalized to different bases, different powers and even negative bases. In
this article we consider generalizations to fractional number bases. Let
be the sum of
the
-th powers
of the digits of
base
.
Let
be the smallest nonnegative integer for which there exists a positive integer
satisfying
. We prove that
such a
, called
the height of
,
exists for all
,
and that, if
or
,
then
can be arbitrarily large.
Keywords
happy numbers, fractional base, digital representation