\begin{definition}[Definition:Sequence Coding]
Let $\left \langle {a_1, a_2, \ldots, a_k}\right \rangle$ be a finite sequence in $\N^*$ (that is, $\forall i \in \left\{{1, 2, \ldots, k}\right\}: a_i > 0$).
Let $p_i$ be the $i$th prime number, so that:
* $p_1 = 2$
* $p_2 = 3$
* $p_3 = 5$
* $p_4 = 7$
etc.
Let $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ where $p_i$ is the $i$th prime number.
Then $n \in \N$ '''codes the sequence $\left \langle {a_1, a_2, \ldots, a_k}\right \rangle$''', or $n$ is the '''code number for the sequence $\left \langle {a_1, a_2, \ldots, a_k}\right \rangle$'''.
The set of all code numbers of finite sequences in $\N$ is denoted $\operatorname{Seq}$.
Note that $n \in \operatorname{Seq}$ {{iff}} $n$ is divisible by all the primes $p_1, p_2, \ldots, p_k$ , where $p_k$ is the ''largest'' prime dividing $n$.
Category:Definitions/Mathematical Logic
\end{definition}