The question verbatim is as follows:
Show that the length of a sequence is a primitive recursive function of its code if we code sequences $\langle n_0, ..., n_{k - 1} \rangle$ of positive numbers as $p_0^{n_0} \cdot ... \cdot p_{k - 1}^{n_{k - 1} + 1}$ (I assume the $n_{k - 1} + 1$ should be $n_{k - 1}$ and $p_i$ is the $i$'th prime, but this is what it says).
If I understand the question correctly, we need to define a function $f$ such that $f(p_0^{n_0} \cdot ... \cdot p_{k - 1}^{n_{k - 1} + 1}) = k - 1$ by giving $f(0)$ and $f(x + 1)$ in terms of $f(x)$ and primitive recursive functions.
The problems I have with this question is that $0$ is not the code for any sequence since the code is positive and that the prime factorizations of $x$ and $x + 1$ do not look alike at all in general.