Arithmetic functions -- there are some nice, very accessible results about arithmetic functions and some questions about these (notably the sum of divisors, $\sigma$) date back to the Greeks (perfect numbers, amicable pairs, etc.) There is also a nice algebra of arithmetic functions using the operation of convolution. Many basic number-theoretic ideas (RSA!) have arithmetic functions hiding in the background.
Example of arithmetic functions include the Euler totient $\phi$-function, $\sigma(n)=$ sum of divisors of $n$, $\tau(n)=$ number of prime divisors of $n$ and the Moebius function $\mu$.
I once had a small team of undergraduates (who had no higher mathematics in their backgrounds) create their own arithmetic functions and analyze questions about iterates of the functions.