  We introduce a modification of Pillai's prime map: the prime-power map. This
map fixes $1$, divides its argument by $p$ if it is a prime-power $p^k$,
otherwise subtracts from its argument the largest prime-power not exceeding it.
We study the iteration of this map over the positive integers, developing,
firstly, results parallel to those known for the prime map. Subsequently, we
compare its dynamical properties to those of a more manageable variant of the
map under which any orbit admits an explicit description. Finally, we present
some experimental observations, based on which we conjecture that almost every
orbit of the prime-power map contains no prime-power.
