[Arithmetic functions][1] -- 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. 


  [1]: http://en.wikipedia.org/wiki/Arithmetic_functions