$$a=\sum_{k=0}^c(k+2)^{s_k}$$ Can every positive integer eventually be expressed in this form where $s_k\ge1$ and $c\ge1$?  
  
My Take:  
First, I'll figure out the conditions for an even/odd number  
Since $k^n$ is odd when odd and even when even, $(2k+1)^n\equiv1$ mod $2$, $(2k)^n\equiv0$ mod $2$.  
From this, you find that for $a$ to be even, $c=4n,4n+1$. Also, for $a$ to be odd, $c=4n-1,4n-2$.  
  
I couldn't find anything other than this, but there is a high chance it is possible to find a proof for this statement.