Find a sequence with an interesting property I was reading an Elon Lages Lima's book ("Curso de Análise", IMPA) and there is an interesting question. Find a function $f$ from $\mathbb N$ onto $\mathbb N$ such that $\forall n\in \mathbb N$, the preimage set $f^{-1}(\{n\})$ is infinite. I don´t have any idea how to construct that.
Let $f(n) = k$ if $n = 2^k \cdot (2l -1)$ for natural numbers $k, l$ -- in other words, if $2^k$ is the highest power of $2$ that divides $n$. Then $f^{-1}(k) = \{2^k \cdot 1, 2^k \cdot 3, 2^k \cdot 5, 2^k \cdot 7\ldots \}$.
How about this:


{x1: x1 is prime OR 1}$\to 1$.
{x2: $x_2=2\cdot p$, for prime p>2}$\to 2$.
{x3: $x_3=3\cdot p$, for prime p>3}$\to 3$.
{x4: $x_4=4\cdot p$, for prime p>4}$\to 4$ 
etc.


Sorry, I was a little sloppy.