Definition:Sequence Coding
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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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