  We define a uniformly behaved in ${\mathbb N}$ arithmetic sequence ${\bf a}$
and an ${\bf a}$-mean Lyapunov stable dynamical system $f$. We consider the
mean partial sum of a continuous function $\phi$ over the ${\bf a}$-orbit of
$f$ up to $N$. The main result we prove in the paper is that the mean partial
sum converges pointwise if ${\bf a}$ is uniformly behaved in ${\mathbb N}$ and
$f$ is minimal and uniquely ergodic and ${\bf a}$-mean Lyapunov stable. In
addition, if ${\bf a}$ is also completely additive, we then prove that the mean
partial sum of a continuous function $\phi$ over the square-free ${\bf
a}$-orbit of $f$ up to $N$ converges pointwise as well. All equicontinuous
dynamical systems are ${\bf a}$-mean Lyapunov stable for any sequence ${\bf
a}$. When ${\bf a}$ is uniformly distributed in ${\mathbb Z}$, we give two
non-trivial examples of ${\bf a}$-mean Lyapunov stable dynamical systems. We
give several examples of uniformly behaved in ${\mathbb N}$ sequences,
including the counting function of the prime factors in natural numbers, the
subsequence of natural numbers indexed by the Thue-Morse (or Rudin-Shapiro)
sequence, and the sequence of even (or odd) prime factor natural numbers. We
also show that the sequence of square-free natural numbers (or even (or odd)
prime factor square-free natural numbers) is rotationally distributed in
${\mathbb N}$ but not uniformly distributed in ${\mathbb Z}$, thus not
uniformly behaved in ${\mathbb N}$. We derive other consequences from the main
result relevant to number theory and ergodic theory/dynamical systems.
