I like Achim Krause's answer, but I prefer a simpler explanation. Here goes:

Place the 2 tape to cover zero, and for $k \gt 0$ place the tape for the $2k$th prime $p_{2k}$ to cover the $k$th positive odd number $(2k-1)$, and place the tape for $p_{2k+1}$ to cover the corresponding negative odd number $(1-2k)$. As $p_n \geq 2n-1$, any number in $[-n,n]$ will have colors only from tapes for $p_k$ with $k+2 \leq n$. So all numbers are colored with only finitely many colors. (That they each get a color is left to the reader.)

This should make apparent ways to extend the result in weak subsystems of arithmetic for sequences $q_n$ replacing $p_n$ that are definable in such a system and are provably increasing in that system.

Gerhard "Looking To Make Things Simpler" Paseman, 2014.04.09

I like Achim Krause's answer, but I prefer a simpler explanation. Here goes:

Place the 2 tape to cover zero, and for $k \gt 0$ place the tape for the $2k$th prime $p_{2k}$ to cover the $k$th positive odd number $(2k-1)$, and place the tape for $p_{2k+1}$ to cover the corresponding negative odd number $(1-2k)$. As $p_n \geq 2n-1$, any number in $[-n,n]$ will have colors only from tapes for $p_k$ with $k \leq n + 2$. So all numbers are colored with only finitely many colors. (That they each get a color is left to the reader.)

This should make apparent ways to extend the result in weak subsystems of arithmetic for sequences $q_n$ replacing $p_n$ that are definable in such a system and are provably increasing in that system.

Gerhard "Looking To Make Things Simpler" Paseman, 2014.04.09