I'm looking at this map that maps natural numbers to sequences of natural numbers. Take a natural number $n$ and construct the unique set $A=\{k_1,k_2,k_3,...,k_m\}, k_m \in \mathbb{N}$ such that $n=\prod_{a \in A}^{} p_m^{a}$ (I assume no trailing zeroes to allow A to be unique). For example, n=1 maps to $\{\}$, n=2 maps to $\{1\}$, n=3 maps to $\{0,1\}$,n=4 maps to $\{2\}$ etc. This map takes care of any finitely long sequence.
It seems like it also covers all countably infinitely long sequences as well. Assume that the longest sequence is only of length k. Then that implies that there are only k distinct primes in the natural numbers, which is false. Since the length of the sequences cannot be bounded, it seems like sequences of countably infinite should be covered by this map too.
But I know this isn't right, since it shouldn't be possible to map the natural numbers into all sequences of natural numbers. What is wrong with my understanding of this map? Thank you for your time!