Definition:Sequence Coding

Definition

Let $\sequence {a_1, a_2, \ldots, a_k}$ be a finite sequence in $\N_{>0}$ (that is, $\forall i \in \set {1, 2, \ldots, k}: a_i > 0$).

Let $p_i$ be the $i$th prime number, so that:

\(\ds p_1\)

\(=\)

\(\ds 2\)

\(\ds p_2\)

\(=\)

\(\ds 3\)

\(\ds p_3\)

\(=\)

\(\ds 5\)

\(\ds p_4\)

\(=\)

\(\ds 7\)

and so on.

Let $n = {p_1}^{a_1} {p_2}^{a_2} \cdots {p_k}^{a_k}$.

Then $n \in \N$ codes the sequence $\sequence {a_1, a_2, \ldots, a_k}$, or $n$ is the code number for the sequence $\sequence {a_1, a_2, \ldots, a_k}$.

The set of all code numbers of finite sequences in $\N$ is denoted $\operatorname{Seq}$.

Note that $n \in \operatorname{Seq}$ if and only if $n$ is divisible by all the primes $p_1, p_2, \ldots, p_k$ , where $p_k$ is the largest prime which is a divisor of $n$.

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